Weak Inequality of Integers iff Strict Inequality with Integer plus One

Let $a, b \in \Z$ be integers. {{TFAE}} :$(1): \quad a \le b$ :$(2): \quad a < b + 1$ where: :$\le$ is the ordering on the integers :$<$ is the strict ordering on the integers.


Set of Orbits forms Partition

Let $G$ be a group. Let $X$ be a set. Let $G$ act on $X$. Then the set of orbits of the group action forms a partition of $X$.


Interior of Closure is Regular Open

Let $T = \left({S, \tau}\right)$ be a topological space. Let $H \subseteq S$. Then $H^{- \circ}$ is regular open.


Jordan's Lemma

Consider a complex-valued, continuous function $f$ defined on the contour: :$C_r = \left\{r e^{i \theta}: 0 \le \theta \le \pi \right\}, \ r > 0$ If the function $f$ is of the form: :$f \left({z}\right) = e^{i a z} g \left({z}\right), \ a > 0, \ z \in C_r$ Then: :$\displaystyle \left\vert{\int_{C_r} f \left({z}\right) \rd z}\right\vert \le \frac \pi a \max_{0 \le \theta \le \pi} \left\vert{g \left(re^{i \theta}\right)}\right\vert$


Lobachevsky Integral Formula

Let $f$ be a continuous function, periodic in $\pi$. Then: :$\displaystyle \int_0^\infty \frac {\sin x} x f \left({x}\right) \rd x = \int_0^{\frac \pi 2} f \left({x}\right) \rd x$


Dirichlet Series of Convolution of Arithmetic Functions

Let $f, g: \N \to \C$ be arithmetic functions. Let $h = f * g$ be their Dirichlet convolution. Let $F, G, H$ be their Dirichlet series. Let $s$ be a complex number such that $\map F s$ and $\map G s$ converge absolutely. Then $\map H s$ converges absolutely to $\map F s \times \map G s$.


Riemann Zeta Function in terms of Dirichlet Eta Function

Let $\zeta$ be the Riemann zeta function. Let $\eta$ be the Dirichlet eta function. Let $s \in \C$ be a complex number with real part $\sigma > 1$. Then $\zeta(s) = \dfrac 1 {1 - 2^{1-s}} \eta \left({s}\right)$.


Functional Equation for Riemann Zeta Function

Let $\zeta$ be the Riemann zeta function. Let $\map \zeta s$ have an analytic continuation for $\map \Re s > 0$. Then: :$\displaystyle \map \Gamma {\frac s 2} \pi^{-s/2} \map \zeta s = \map \Gamma {\frac {1 - s} 2} \pi^{\frac {s - 1} 2} \map \zeta {1 - s}$ where $\Gamma$ is the gamma function


Integral Representation of Riemann Zeta Function in terms of Fractional Part

Let $\zeta$ be the Riemann zeta function. Let $s\in\C$ be a complex number with real part $\sigma>1$. Then  :$\displaystyle \zeta \left({s}\right) = \frac s {s - 1} - s \int_1^\infty \left\{ {x}\right\} x^{-s - 1} \rd x$ where $\left\{{x}\right\}$ denotes the fractional part of $x$.


Binomial Form of Relation between Riemann Zeta Function and Dirichlet Eta Function

Let $\zeta$ be the Riemann zeta function. Let $s\in\C$ be a complex number with real part $\Re(s)>1$. Then: $\displaystyle \zeta \left({s}\right) = \frac 1 {1 - 2^{1-s} } \sum_{n \mathop = 0}^\infty \left({\frac 1 {2^{n+1} } \sum_{k \mathop = 0}^n \left({-1}\right)^k \binom n k \left({k + 1}\right)^{-s} }\right)$


Analytic Continuations of Riemann Zeta Function to Complex Plane

The Riemann zeta function $\zeta$ has a unique analytic continuation to $\C\setminus\{1\}$.


Analytic Continuations of Riemann Zeta Function to Right Half-Plane

The Riemann zeta function has a unique analytic continuation to $\{s \in \C : \Re(s) > 0\}\setminus\{1\}$, the half-plane $\Re(s)>0$ minus the point $s=1$.


Dirichlet Series of Inverse of Arithmetic Function

Let $f : \N \to\C$ be an arithmetic function. Let $g : \N \to \C$ be an Dirichlet inverse of $f$. Let $F, G$ be their Dirichlet series. Let $s \in \C$ such that both $F(s)$ and $G(s)$ converge absolutely. Then $F(s) \cdot G(s) = 1$.


Invertibility of Arithmetic Functions

Let $f: \N \to \C$ be an arithmetic function. Then $f$ has a Dirichlet inverse {{iff}}: :$\map f 1 \ne 0$


Equivalence of Definitions of Consistent Set of Formulas

{{TFAE|def = Consistent (Logic)/Proof System/Propositional Logic|view = Consistent Proof System for Propositional Logic}} Let $\LL_0$ be the language of propositional logic. Let $\mathscr P$ be a proof system for $\LL_0$. Let $\FF$ be a collection of logical formulas.


Analytic Continuation of Riemann Zeta Function using Mellin Transform of Fractional Part

Let $\zeta$ denote the Riemann zeta function. The analytic continuation of $\zeta$ to the half-plane $\map \Re s > 0$ is given by: :$\displaystyle \frac s {s - 1} - s \int_1^\infty \fractpart x x^{-s - 1} \rd x$ where $x^{-s - 1}$ takes the principle value $e^{-\map \ln x \paren {s + 1} }$


Analytic Continuation of Riemann Zeta Function using Dirichlet Eta Function

Let $\zeta$ be the Riemann zeta function. Let $\eta$ be the Dirichlet Eta Function. Then: :$\dfrac 1 {1 - 2^{1 - s} } \map \eta s$ defines an analytic continuation of $\zeta$ to the half-plane $\map \Re s > 0$ minus $s = 1$.


Analytic Continuation of Riemann Zeta Function using Jacobi Theta Function

Let $\zeta$ be the Riemann zeta function. Then :$\displaystyle\frac{\pi^{s/2}}{\Gamma \left({\frac s 2}\right)} \cdot\left( - \frac 1 {s \left({1 - s}\right)} + \int_1^\infty \left({x^{s / 2 - 1} + x^{- \left({s + 1}\right) / 2} }\right) \omega \left({x}\right) \ \mathrm d x \right)$ defines an analytic continuation of $\zeta$ to the half-plane $\Re(s)>0$ minus $s=1$.


Complex Modulus of Real Number equals Absolute Value

Let $x \in \R$ be a real number. Then the complex modulus of $x$ equals the absolute value of $x$.


Equivalence of Definitions of Ceiling Function

Let $x$ be a real number. {{TFAE|def = Ceiling Function}}


Characterizing Property of Infimum of Subset of Real Numbers

Let $S \subset \R$ be a non-empty subset of the real numbers. Let $S$ be bounded below. Let $\alpha \in \R$. {{TFAE}} :$(1): \quad \alpha$ is the infimum of $S$ :$(2): \quad \alpha$ is a lower bound for $S$ ::and: :::$\forall \epsilon \in \R_{> 0}$ there exists $x \in S$ with $x < \alpha + \epsilon$


Characterizing Property of Supremum of Subset of Real Numbers

Let $S \subset \R$ be a non-empty subset of the real numbers. Let $S$ be bounded above. Let $\omega \in \R$. {{TFAE}} :$(1): \quad \omega$ is the supremum of $S$ :$(2): \quad \omega$ is an upper bound for $S$ ::and: :::$\forall \epsilon \in \R_{> 0}$ there exists $x \in S$ with $x > \omega - \epsilon$


Limit of Positive Real Sequence is Positive

Let $\sequence {x_n}$ be a sequence of positive real numbers. Let $x_n$ converge to $L$. Then $L \ge 0$.


Real Sequence with Nonzero Limit is Eventually Nonzero

Let $\sequence {x_n}$ be a real sequence. Let $\sequence {x_n}$ converge to $a \ne 0$. Then: :$\exists N \in \N: \forall n \ge N: x_n \ne 0$ That is, eventually every term of $\sequence {x_n}$ becomes non-zero.


Mittag-Leffler Expansion for Cotangent Function

:$\displaystyle \pi \cot \pi z = \frac 1 z + 2 \sum_{n \mathop = 1}^\infty \frac z {z^2 - n^2}$ where: :$z \in \C$ is not an integer :$\cot$ is the cotangent function.


Laurent Series Expansion for Cotangent Function

:$\displaystyle \pi \cot \pi z = \frac 1 z - 2 \sum_{n \mathop = 1}^\infty \map \zeta {2 n} z^{2 n - 1}$ where: :$z \in \C$ such that $\cmod z < 1$ :$\zeta$ is the Riemann Zeta function.


Equivalence of Definitions of Removable Discontinuity of Real Function

Let $A \subseteq \R$ be a subset of the real numbers. Let $f : A \to \R$ be a real function. Let $f$ be discontinuous at $a\in A$. {{TFAE|def = Removable Discontinuity of Real Function|view = removable discontinuity}}


Equivalence of Definitions of Polynomial Function on Subset of Ring

Let $R$ be a commutative ring with unity. Let $S \subset R$ be a subset. {{TFAE|def = Polynomial Function/Ring |view = polynomial function}}


Equivalence of Definitions of Polynomial Ring in Multiple Variables

Let $R$ be a commutative ring with unity. The following definitions of polynomial ring are equivalent in the following sense: : For every two constructions, there exists an $R$-isomorphism which sends indeterminates to indeterminates. {{explain|this statement has to be made more precise}} === Definition 1: As the monoid ring on a free monoid on a set === {{:Definition:Polynomial Ring/Monoid Ring on Free Monoid on Set}} {{ExpandList}}


Monomials form Basis of Polynomial Ring/One Variable

Let $R$ be a commutative ring with unity. Let $R \sqbrk X$ be a polynomial ring over $R$ in the variable $X$. Then the monomials of $R \sqbrk X$ are a basis of $R \sqbrk X$ as a module over $R$.


Equivalence of Definitions of Polynomial in Ring Element

Let $R$ be a commutative ring. Let $S$ be a subring with unity of $R$. Let $x\in R$. {{TFAE|def = Polynomial in Ring Element}}


Equality of Monomials of Polynomial Ring in One Variable

Let $R$ be a commutative ring with unity. Let $R \sqbrk X$ be a polynomial ring in one variable $X$ over $R$. Let $k, l \in \N$ be distinct natural numbers. Then the mononomials $X^k$ and $X^l$ are distinct, where $X^k$ denotes the $k$th power of $X$.


Ring Isomorphic to Polynomial Ring is Polynomial Ring/One Variable

Let $R$ be a commutative ring with unity. Let $R \sqbrk X$ be a polynomial ring in one variable $X$ over $R$. Let $\iota : R \to R \sqbrk X$ denote the canonical embedding. Let $S$ be a commutative ring with unity and $f: R \sqbrk X \to S$ be a ring isomorphism. Then $\struct {S, f \circ \iota, \map f X}$ is a polynomial ring in one variable $\map f X$ over $R$.


Equality of Monomials of Polynomial Ring in Multiple Variables

Let $R$ be a commutative ring with unity. Let $I$ be a set. Let $R \sqbrk {\sequence {x_i}_{i \mathop \in I} }$ be a polynomial ring in $I$ variables $\sequence {x_i}_{i \mathop \in I}$ over $R$. Let $a, b : I \to \N$ be distinct mappings with finite support. Then the monomials $\ds \prod_{i \mathop \in I} X_i^{a_i}$ and $\ds \prod_{i \mathop \in I} X_i^{b_i}$ are distinct, where: :$X_i^k$ denotes the $k$th power of $X_i$ :$\prod$ denotes product with finite support


Mittag-Leffler Expansion for Cosecant Function

:$\displaystyle \pi \cosec \pi z = \frac 1 z + 2\sum_{n \mathop = 1}^\infty \paren {-1}^n \frac z {z^2 - n^2}$


Upper Sum Never Smaller than Lower Sum for any Pair of Subdivisions

Let $\closedint a b$ be a closed real interval. Let $f$ be a bounded real function defined on $\closedint a b$. Let $P$ and $Q$ be finite subdivisions of $\closedint a b$. Let $\map L P$ be the lower sum of $f$ on $\closedint a b$ with respect to $P$. Let $\map U Q$ be the upper sum of $f$ on $\closedint a b$ with respect to $Q$. Then $\map L P \le \map U Q$.


Set Finite iff Surjection from Initial Segment of Natural Numbers

Let $S$ be a set. Then $S$ is finite {{iff}} for some $n \in \N$ there exists a surjection $f: \N_{< n} \to S$. Here, $\N_{< n}$ denotes an initial segment of $\N$.


Set Finite iff Injection to Initial Segment of Natural Numbers

Let $S$ be a set. Then $S$ is finite {{iff}} for some $n \in \N$ there exists an injection $f: S \to \N_{< n}$. Here, $\N_{< n}$ denotes an initial segment of $\N$.


Finite Product of Finite Sets is Finite

Let $\sequence {S_n}$ be a sequence of finite sets. Let $\displaystyle \prod_{k \mathop = 1}^n S_k$ be their Cartesian product. Then $\displaystyle \prod_{k \mathop = 1}^n S_k$ is also a finite set.


Existence of Integers with Multiplicative Persistence Greater than 11

It is not known whether there exists a number $n$ such that: :$P \left({n}\right) = 12$ where $P \left({n}\right)$ denotes the multiplicative persistence of $n$.


Smallest Strong Fibonacci Pseudoprime of Type I

The smallest strong Fibonacci pseudoprime of type I is $443 \, 372 \, 888 \, 629 \, 441$.


Infimum of Upper Sums Never Smaller than Lower Sum

:$\inf_P \map U P \ge \map L S$


Supremum of Lower Sums Never Greater than Upper Sum

:$\sup_P \map L P \le \map U S$


Properties of 5,559,060,566,555,523

$3^{33} = 5 \, 559 \, 060 \, 566 \, 555 \, 523$ has the following properties: : It has a remarkably large number of $5$s (half of its digits). : Multiply it by $2$, $4$ or $6$, and the result has $10$ of a particular digit. : Multiply it by $8$, and $9$ of the digits of the result are $4$.


First Occurrence of Prime Gap of 864

The first occurrence of a prime gap of $864$ is between $6 \, 505 \, 941 \, 701 \, 960 \, 039$ and $6 \, 505 \, 941 \, 701 \, 960 \, 903$.


Hardy-Ramanujan Number/Examples/48,988,659,276,962,496

The $5$th Hardy-Ramanujan number $\map {\operatorname {Ta} } 5$ is $48 \, 988 \, 659 \, 276 \, 962 \, 496$: {{begin-eqn}} {{eqn | l = 48 \, 988 \, 659 \, 276 \, 962 \, 496       | r = 38 \, 787^3 + 365 \, 757^3       | c =  }} {{eqn | r = 107 \, 839^3 + 362 \, 753^3       | c =  }} {{eqn | r = 205 \, 292^3 + 342 \, 952^3       | c =  }} {{eqn | r = 221 \, 424^3 + 336 \, 588^3       | c =  }} {{eqn | r = 231 \, 518^3 + 331 \, 954^3       | c =  }} {{end-eqn}}


Reciprocal of 19 from Sum of Powers of 2 Backwards

The decimal expansion of the reciprocal of $19$ can be constructed by summing the powers of $2$, offset progressively backwards by $1$ digit: <pre>                          1                         2                        4                       8                     16                    32                   64                 128                256               512             1024            2048           4096          8192        16384       32768      65536    131072 + 262144  ...... -------------------------- .......1052631578947368421 </pre> while: :$\dfrac 1 {19} = 0 \cdotp \dot 05263 \, 15789 \, 47368 \, 42 \dot 1$


Multiples of Reciprocals with Maximum Period form Magic Square

Let $p$ be a prime number whose reciprocal has a decimal expansion which has the maximum period, that is, $p - 1$. Let the first $p - 1$ digits of $\dfrac 1 p$ be expressed as an integer $n$ (with one or more leading zeroes as it is presented). Then the first $p - 1$ multiples of $n$, when listed in order of size, arrange themselves as a magic square such that: :the sum of the elements of each row :the sum of the elements in each column are the same.


General Fibonacci Sequence whose Terms are all Composite

The general Fibonacci sequence $\left\langle{a_n}\right\rangle$ defined as: :$a_n = \begin{cases} r & : n = 0 \\ s & : n = 1 \\ a_{n - 2} + a_{n - 1} & : n > 1 \end{cases}$ where: :$r = 62 \, 638 \, 280 \, 004 \, 239 \, 857$ :$s = 49 \, 463 \, 435 \, 743 \, 205 \, 655$ is such that: : $r$ and $s$ are coprime : all its terms are composite.


Superset of Infinite Set is Infinite

Let $S$ be an infinite set. Let $T \supseteq S$ be a superset of $S$. Then $T$ is also infinite.


Prime Decomposition of 2^58+1

The number $2^{58} + 1$ has the prime decomposition: :$2^{58} + 1 = 5 \times 107 \, 367 \, 629 \times 536 \, 903 \, 681$


Sum of Squares as Product of Factors with Square Roots

:$x^2 + y^2 = \left({x + \sqrt {2 x y} + y}\right) \left({x - \sqrt {2 x y} + y}\right)$


Prime Decomposition of Repunits

The prime decomposition of the repunits is as follows: {{begin-eqn}} {{eqn | l = R_2       | o =        | r = 11       | c =  }} {{eqn | l = R_3       | r = 3 \times 37       | c =  }} {{eqn | l = R_4       | r = 11 \times 101       | c =  }} {{eqn | l = R_5       | r = 41 \times 271       | c =  }} {{eqn | l = R_6       | r = 11 \times 111 \times 91       | c =  }} {{eqn | r = 3 \times 7 \times 11 \times 13 \times 37       | c =  }} {{eqn | l = R_7       | r = 239 \times 4649       | c =  }} {{eqn | l = R_8       | r = 1111 \times 10 \, 001       | c =  }} {{eqn | r = 11 \times 73 \times 101 \times 137       | c =  }} {{eqn | l = R_9       | r = 111 \times 1 \, 001 \, 001       | c =  }} {{eqn | r = 3^2 \times 37 \times 333 \, 667       | c =  }} {{eqn | l = R_{10}       | r = 11 \times 11 \, 111 \times 9091       | c =  }} {{eqn | r = 11 \times 41 \times 271 \times 9091       | c =  }} {{eqn | l = R_{11}       | r = 21 \, 649 \times 513 \, 239       | c =  }} {{eqn | l = R_{12}       | r = 111 \times 1111 \times 900 \, 991       | c =  }} {{eqn | r = 3 \times 7 \times 11 \times 13 \times 37 \times 101 \times 9901       | c =  }} {{eqn | l = R_{13}       | r = 53 \times 79 \times 265 \, 371 \, 653       | c =  }} {{eqn | l = R_{14}       | r = 11 \times 1 \, 111 \, 111 \times 909 \, 091       | c =  }} {{eqn | r = 11 \times 239 \times 4649 \times 909 \, 091       | c =  }} {{eqn | l = R_{15}       | r = 111 \times 11 \, 111 \times 90 \, 090 \, 991       | c =  }} {{eqn | r = 3 \times 31 \times 37 \times 41 \times 271 \times 2 \, 906 \, 161       | c =  }} {{eqn | l = R_{16}       | r = 11 \, 111 \, 111 \times 100 \, 000 \, 001       | c =  }} {{eqn | r = 11 \times 17 \times 73 \times 101 \times 137 \times 5 \, 882 \, 353       | c =  }} {{eqn | l = R_{17}       | r = 2 \, 071 \, 723 \times 5 \, 363 \, 222 \, 357       | c =  }} {{eqn | l = R_{18}       | r = 111 \, 111 \, 111 \times 1 \, 000 \, 000 \, 001       | c =  }} {{eqn | r = 3^2 \times 7 \times 11 \times 13 \times 19 \times 37 \times 52 \, 579 \times 333 \, 667       | c =  }} {{eqn | l = R_{19}       | r = 1 \, 111 \, 111 \, 111 \, 111 \, 111 \, 111       | c =  }} {{eqn | l = R_{20}       | r = 1111 \times 11111 \times 900 \, 099 \, 009 \, 991       | c =  }} {{eqn | r = 11 \times 41 \times 101 \times 271 \times 3541 \times 9091 \times 27 \, 961       | c =  }} {{end-eqn}} {{expand|Go as far as we want}}


Repunit cannot be Square

A repunit (apart from the trivial $1$) cannot be a square.


Second Principle of Recursive Definition

Let $\N$ be the natural numbers. Let $T$ be a set. Let $a \in T$. For each $n \in \N_{>0}$, let $G_n: T^n \to T$ be a mapping. Then there exists exactly one mapping $f: \N \to T$ such that: :$\forall x \in \N: f(x) = \begin{cases} a & : x = 0\\ G_n \left({f(0), \ldots, f(n)}\right) & : x = n + 1 \end{cases}$


Repunit cannot be Cube

A repunit (apart from the trivial $1$) cannot be a cube.


Condition for Repunits to be Coprime

Let $R_p$ and $R_q$ be repunit numbers with $p$ and $q$ digits respectively. Then $R_p$ and $R_q$ are coprime {{iff}} $p$ and $q$ are coprime.


Divisors of Repunit with Composite Index

Let $R_n$ be a repunit number with $n$ digits. Let $n$ be composite such that $n = r s$ where $1 < r < n$ and $1 < s < n$. Then $R_r$ and $R_s$ are both divisors of $R_n$.


Square of Small Repunit is Palindromic

The squares of repunits with up to $9$ digits are palindromic.


Square Number ending in 9 Digits in Reverse Order

The next integer after $1 \, 111 \, 111 \, 111$ whose square ends in $987 \, 654 \, 321$ is: :$2 \, 380 \, 642 \, 361^2 = 5 \, 667 \, 458 \, 050 \, 987 \, 654 \, 321$


5 Numbers such that Sum of any 3 is Square

This set of $5$ integers has the property that the sum of any $3$ of them is square: {{begin-eqn}} {{eqn | l = 26 \, 072 \, 323 \, 311 \, 568 \, 661 \, 931       | o =  }} {{eqn | l = 43 \, 744 \, 839 \, 742 \, 282 \, 591 \, 947       | o =  }} {{eqn | l = 118 \, 132 \, 654 \, 413 \, 675 \, 138 \, 222       | o =  }} {{eqn | l = 186 \, 378 \, 732 \, 807 \, 587 \, 076 \, 747       | o =  }} {{eqn | l = 519 \, 650 \, 114 \, 814 \, 905 \, 002 \, 347       | o =  }} {{end-eqn}}


Total Number of Reachable Positions on Rubik's Cube

The total number $N$ of reachable positions on Rubik's cube is: {{begin-eqn}} {{eqn | l = N       | r = 43 \, 252 \, 003 \, 274 \, 489 \, 856 \, 000       | c =  }} {{eqn | r = \dfrac {8! \times 12! \times 3^8 \times 2^{12} } {2 \times 3 \times 3}       | c =  }} {{end-eqn}}


Smallest Number which is Multiplied by 99 by Appending 1 to Each End

The smallest positive integer which is multiplied by $99$ when $1$ is appended to each end is: :$112 \, 359 \, 550 \, 561 \, 797  \, 732 \, 809$


Smallest Multiply Perfect Number of Order 6

The number $154 \, 345 \, 556 \, 085 \, 770 \, 649 \, 600$ is multiply perfect of order $6$: {{begin-eqn}} {{eqn | l = \sigma \left({154 \, 345 \, 556 \, 085 \, 770 \, 649 \, 600}\right)       | r = 926 \, 073 \, 336 \, 514 \, 623 \, 897 \, 600       | c =  }} {{eqn | r = 6 \times 154 \, 345 \, 556 \, 085 \, 770 \, 649 \, 600       | c =  }} {{end-eqn}} It is the smallest positive integer to be so.


Sequence of 9 Consecutive Integers each with 48 Divisors

The $9$ integers beginning $17 \, 796 \, 126 \, 877 \, 482 \, 329 \, 126 \, 044$ each has $48$ divisors.


Polydivisible Number/Examples/3,608,528,850,368,400,786,036,725

The largest polydivisible number has $25$ digits: :$3 \, 608 \, 528 \, 850 \, 368 \, 400 \, 786 \, 036 \, 725$


No Polydivisible Number with 26 Digits Exists

There exists no polydivisible number with $26$ digits or more.


Probability of All Players receiving Complete Suit at Bridge

The probability of all $4$ players in a game of Bridge being dealt a complete suit is $1$ in $2 \, 235 \, 197 \, 406 \, 895 \, 366 \, 368 \, 301 \, 560 \, 000$.


Powers of 10 Expressible as Product of 2 Zero-Free Factors

The powers of $10$ which can be expressed as the product of $2$ factors neither of which has a zero in its decimal representation are: {{begin-eqn}} {{eqn | l = 10^1       | r = 2 \times 5       | c =  }} {{eqn | l = 10^2       | r = 4 \times 25 }} {{eqn | l = 10^3       | r = 8 \times 125 }} {{eqn | l = 10^4       | r = 16 \times 625 }} {{eqn | l = 10^5       | r = 32 \times 3125 }} {{eqn | l = 10^6       | r = 64 \times 15 \, 625 }} {{eqn | l = 10^7       | r = 128 \times 78 \, 125 }} {{eqn | l = 10^9       | r = 512 \times 1 \, 953 \, 125 }} {{eqn | l = 10^{18}       | r = 262 \, 144 \times 3 \, 814 \, 697 \, 265 \, 625 }} {{eqn | l = 10^{33}       | r = 8 \, 589 \, 934 \, 592 \times 116 \, 415 \, 321 \, 826 \, 934 \, 814 \, 453 \, 125 }} {{end-eqn}}


Sequence of Palindromic Sophie Germain Primes

The number $N = 191 \, 918 \, 080 \, 818 \, 091 \, 909 \, 090 \, 909 \, 190 \, 818 \, 080 \, 819 \, 191$ has the property that: :$N$ is a palindromic Sophie Germain prime :$2 N + 1$ is also a palindromic Sophie Germain prime :$2 \left({2 N + 1}\right) + 1$ is also a palindromic prime, but not a Sophie Germain prime.


Power of 2 containing no Digit 2

$2^{168}$ contains no $2$ anywhere in its decimal representation.


Order of Fischer-Griess Monster

The order of the Fischer-Griess Monster $\mathrm M$ is given as: {{begin-eqn}} {{eqn | l = \order {\mathrm M}       | r = 2^{46} \times 3^{20} \times 5^9 \times 7^6 \times 11^2 \times 13^3 \times 17 \times 19 \times 23 \times 29 \times 31 \times 41 \times 47 \times 59 \times 71       | c =  }} {{eqn | r = 808 \, 017 \, 424 \, 794 \, 512 \, 875 \, 886 \, 459 \, 904 \, 961 \, 710 \, 757 \, 005 \, 754 \, 368 \, 000 \, 000 \, 000       | c =  }} {{end-eqn}}


Smallest Cube whose Sum of Divisors is Cube

The smallest cube $N$ such that $\map \sigma N$ is also a cube is: :$27 \, 418 \, 521 \, 963 \, 671 \, 501 \, 273 \, 905 \, 190 \, 135 \, 082 \, 692 \, 041 \, 730 \, 405 \, 303 \, 870 \, 249 \, 023 \, 209$ where $\map \sigma N$ denotes the $\sigma$ function of $N$: the sum of the divisors of $N$


Uniqueness of Polynomial Ring in One Variable

Let $R$ be a commutative ring with unity. Let $\struct {R \sqbrk X, \iota, X}$ and $\struct {R \sqbrk Y, \kappa, Y}$ be polynomial rings in one variable over $R$. Then there exists a unique ring homomorphism $f: R \sqbrk X \to R \sqbrk Y$ such that: :$f \circ \iota = \kappa$ :$\map f X = Y$ and it is an isomorphism.


Monomials of Polynomial Ring are Linearly Independent/One Variable

Let $R$ be a commutative ring with unity. Let $R \sqbrk X$ be a polynomial ring in one variable $X$ over $R$. Then the set of monomials $\set {X^k : k \in \N}$ is linearly independent.


Equivalence of Definitions of Constant Polynomial

Let $R$ be a commutative ring with unity. Let $P\in R[x]$ be a polynomial in one variable over $R$. {{TFAE|def = Constant Polynomial}}


Algebraic Element of Field Extension is Root of Unique Monic Polynomial of Minimal Degree

Let $L / K$ be a field extension. Let $\alpha \in L$ be algebraic over $K$. Then there exists a unique monic polynomial $f \in K \sqbrk x$ of smallest degree such that $\map f \alpha = 0$, called the '''minimal polynomial'''.


Algebraic Element of Field Extension is Root of Unique Monic Irreducible Polynomial

Let $L/K$ be a field extension. Let $\alpha \in L$ be algebraic over $K$. Then there exists a unique irreducible monic polynomial $f\in K[x]$ such that $f(\alpha) = 0$, called the '''minimal polynomial'''.


Annihilating Polynomial of Minimal Degree is Irreducible

Let $L / K$ be a field extension. Let $\alpha \in L$ be algebraic over $K$. Let $f \in K[x]$ be a nonzero polynomial of minimal degree such that $\map f \alpha = 0$. Then $f$ is irreducible in $K \sqbrk x$.


Nonzero Ideal of Polynomial Ring over Field has Unique Monic Generator

Let $K$ be a field. Let $K \sqbrk x$ be the polynomial ring in one variable over $K$. Let $I \subseteq K \sqbrk x$ be a nonzero ideal. Then $I$ is generated by a unique monic polynomial.


Equivalence of Definitions of Integral Element of Algebra

{{TFAE|def = Integral Element of Algebra}} Let $A$ be a commutative ring with unity. Let $f : A \to B$ be a commutative $A$-algebra. Let $b\in B$.


Product of Closed Sets is Closed

Let $\family {\struct {S_i, \tau_i} }_{i \mathop \in I}$ be a family of topological spaces where $I$ is an arbitrary indexing set. Let $\displaystyle S = \prod_{i \mathop \in I} S_i$. Let $T = \struct {S, \TT}$ be the product space of $\family {\struct {S_i, \tau_i} }_{i \mathop \in I}$ with Tychonoff topology $\TT$. Suppose we have an indexed family of sets $\family {C_i}_{i \mathop \in I}$, where each $C_i$ is closed in $\struct {S_i, \tau_i}$. Then $\displaystyle \prod_{i \mathop \in I} C_i$ is closed in $\struct {S, \TT}$.


Prime Decomposition of 9th Fermat Number

The prime decomposition of the $9$th Fermat number is given by: {{begin-eqn}} {{eqn | l = 2^{\left({2^9}\right)} + 1       | r = 13 \, 407 \, 807 \, 929 \, 942 \, 597 \, 099 \, 574 \, 024 \, 998 \, 205 \, 846 \, 127 \, 479 \, 365 \, 820 \, 592 \, 393 \, 377 \, 723 \, 561 \, 443 \, 721 \, 764 \, 030 \, 073 \, 546 \, 976 \, 801 \, 874 \, 298 \, 166 \, 903 \, 427 \, 690 \, 031 \, 858 \, 186 \, 486 \, 050 \, 853 \, 753 \, 882 \, 811 \, 946 \, 569 \, 946 \, 433 \, 649 \, 006 \, 084 \, 097       | c =  }} {{eqn | r = 2 \, 424 \, 833       | c =  }} {{eqn | o =       | ro= \times       | r = 7 \, 455 \, 602 \, 825 \, 647 \, 884 \, 208 \, 337 \, 395 \, 736 \, 200 \, 454 \, 918 \, 783 \, 366 \, 342 \, 657       | c =  }} {{eqn | o =       | ro= \times       | r = 741 \, 640 \, 062 \, 627 \, 530 \, 801 \, 524 \, 787 \, 141 \, 901 \, 937 \, 474 \, 059 \, 940 \, 781 \, 097 \, 519 \, 023 \, 905 \, 821 \, 316 \, 144 \, 415 \, 759 \, 504 \, 705 \, 008 \, 092 \, 818 \, 711 \, 693 \, 940 \, 737       | c =  }} {{eqn | r = \paren {2^5 \times 37 \times 2^{11} + 1}       | c =  }} {{eqn | o =       | ro= \times       | r = \paren {19 \times 47 \times 82 \, 488 \, 781 \times 1 \, 143 \, 290 \, 228 \, 161 \, 321 \times 43 \, 226 \, 490 \, 359 \, 557 \, 706 \, 629 \times 2^{11} + 1}       | c =  }} {{eqn | o =       | ro= \times       | r = \paren {1129 \times 26 \, 813 \times 40 \, 644 \, 377 \times 17 \, 338 \, 437 \, 577 \, 121 \times 16 \, 975 \, 143 \, 302 \, 271 \, 505 \, 426 \, 897 \, 585 \, 653 \, 131 \, 126 \, 520 \, 182 \, 328 \, 037 \, 821 \, 729 \, 720 \, 833 \, 840 \, 187 \, 223 \times 2^{11} + 1}       | c =  }} {{end-eqn}}


Prime Number Formed by Concatenating Consecutive Integers down to 1

Let $N$ be an integer whose decimal representation consists of the concatenation of all the integers from a given $n$ in descending order down to $1$. Let the $N$ that is so formed be prime. The only $n$ less than $100$ for which this is true is $82$. That is: :$82 \, 818 \, 079 \, 787 \, 776 \ldots 121 \, 110 \, 987 \, 654 \, 321$ is the only prime number formed this way starting at $100$ or less.


Yoneda Lemma for Covariant Functors

Let $C$ be a locally small category. Let $\mathbf{Set}$ be the category of sets. === Bijection === {{:Bijection in Yoneda Lemma for Covariant Functors}} === Naturality === {{:Naturality of Yoneda Lemma for Covariant Functors}}


Naturality of Yoneda Lemma for Covariant Functors

Let $[C, \mathbf{Set}]$ be the covariant functor category. Let $C \times [C, \mathbf{Set}] $ be the product category. Let $C \times [C, \mathbf{Set}] \to \mathbf{Set} : (A, F) \mapsto \operatorname{Nat}(h^A, F)$ be the covariant functor defined as the composition of the hom bifunctor and the product of the contravariant Yoneda functor $h^-$ and the identity functor $\operatorname{id}_{[C, \mathbf{Set}]}$. Let $\operatorname{ev} : C \times [C, \mathbf{Set}] \to \mathbf{Set} : (A, F) \mapsto F(A)$ be the functor evaluation functor. Then $\Phi_{(A, F)} : \operatorname{Nat}(h^A, F) \to F(A) : \eta \mapsto \eta_A(\operatorname{id}_A)$ defines a natural isomorphism, where $\operatorname{id}_A$ is the identity morphism of $A$.


Alternating Even-Odd Digit Palindromic Prime

Let the notation $\left({abc}\right)_n$ be interpreted to mean $n$ consecutive repetitions of a string of digits $abc$ concatenated in the decimal representation of an integer. The integer: :$\left({10987654321234567890}\right)_{42} 1$ has the following properties: : it is a palindromic prime with $841$ digits : its digits are alternately odd and even.


Bijection in Yoneda Lemma for Covariant Functors

Let $F: C \to \mathbf {Set}$ be a covariant functor. Let $A \in C$ be an object. Let $I_A$ be its identity morphism. Let $h^A = \map {\operatorname {Hom} } {A, -}$ be its covariant hom-functor. The class of natural transformations $\map {\operatorname {Nat} } {h^A, F}$ is a small class, and: :$\alpha: \map {\operatorname {Nat} } {h^A, F} \to \map F A: \eta \mapsto \map {\eta_A} {I_A}$ :$\beta: \map F A \to \map {\operatorname {Nat} } {h^A, F}: u \mapsto \paren {X \mapsto \paren {f \mapsto \map {\paren {\map F f} } u} }$ are inverses of each other.


Yoneda Lemma for Contravariant Functors

Let $C$ be a locally small category. Let $\mathbf{Set}$ be the category of sets. === Bijection === {{:Bijection in Yoneda Lemma for Contravariant Functors}} === Naturality === {{:Naturality of Yoneda Lemma for Contravariant Functors}}


Naturality of Yoneda Lemma for Contravariant Functors

Let $[C^{\operatorname{op}}, \mathbf{Set}]$ be the contravariant functor category. Let $C^{\operatorname{op}} \times [C^{\operatorname{op}}, \mathbf{Set}] $ be the product category. Let $C^{\operatorname{op}} \times [C^{\operatorname{op}}, \mathbf{Set}] \to \mathbf{Set} : (A, F) \mapsto \operatorname{Nat}(h_A, F)$ be the covariant functor defined as the composition of the hom bifunctor and the product of the opposite of the covariant Yoneda functor $h_-$ and the identity functor $\operatorname{id}_{[C^{\operatorname{op}}, \mathbf{Set}]}$. Let $\operatorname{ev} : C^{\operatorname{op}} \times [C^{\operatorname{op}}, \mathbf{Set}] \to \mathbf{Set} : (A, F) \mapsto F(A)$ be the contravariant functor evaluation functor. Then $\Phi_{(A, F)} : \operatorname{Nat}(h_A, F) \to F(A) : \eta \mapsto \eta_A(\operatorname{id}_A)$ defines a natural isomorphism, where $\operatorname{id}_A$ is the identity morphism of $A$.


Period of Reciprocal of Repunit 1031 is 1031

The decimal expansion of the reciprocal of the repunit prime $R_{1031}$ has a period of $1031$. :$\dfrac 1 {R_{1031}} = 0 \cdotp \underbrace{\dot 000 \ldots 000}_{1030} \dot 9$ This is the only prime number to have a period of exactly $1031$.


Titanic Prime whose Digits are all Prime

The integer defined as: :$7532 \times \dfrac {10^{1104} - 1} {10^4 - 1} + 1$ is a titanic prime all of whose digits are themselves prime. That is: :$\underbrace{7532}_{275} 7533$


Titanic Prime whose Digits are all 0 or 1

The integer defined as: :$10^{641} \times \dfrac{10^{640} - 1} 9 + 1$ is a titanic prime all of whose digits are either $0$ or $1$. That is: :$\left({1}\right)_{640} \left({0}\right)_{640} 1$ where $\left({a}\right)_b$ means $b$ instances of $a$ in a string.


Titanic Sophie Germain Prime

The integer defined as: :$39 \, 051 \times 2^{6001} - 1$ is a titanic prime which is also a Sophie Germain prime: {{begin-eqn}} {{eqn | o =        | r = 11820 \, 50794 \, 19125 \, 52383 \, 74423 \, 53078 \, 56017 \, 05024 \, 84819 \, 01689 }} {{eqn | o =        | r = 74975 \, 95139 \, 68621 \, 89553 \, 48654 \, 81137 \, 72841 \, 27658 \, 52217 \, 40999 }} {{eqn | o =        | r = 04778 \, 71896 \, 78015 \, 63535 \, 94741 \, 82340 \, 68638 \, 88011 \, 18130 \, 14219 }} {{eqn | o =        | r = 81435 \, 50235 \, 73607 \, 51980 \, 74200 \, 04306 \, 58030 \, 53360 \, 79821 \, 16678 }} {{eqn | o =        | r = 32541 \, 21729 \, 72493 \, 53731 \, 27605 \, 59447 \, 95967 \, 46064 \, 11137 \, 07858 }} {{eqn | o =        | r = 37078 \, 27755 \, 33462 \, 32179 \, 66482 \, 80947 \, 33386 \, 65681 \, 87582 \, 11189 }} {{eqn | o =        | r = 25630 \, 83169 \, 50526 \, 70023 \, 66301 \, 83449 \, 99960 \, 25913 \, 90035 \, 61496 }} {{eqn | o =        | r = 03726 \, 62661 \, 50693 \, 56343 \, 90085 \, 30468 \, 46645 \, 69888 \, 03202 \, 50070 }} {{eqn | o =        | r = 38139 \, 19172 \, 69637 \, 71838 \, 13812 \, 48256 \, 38384 \, 37787 \, 83423 \, 06357 }} {{eqn | o =        | r = 09062 \, 96393 \, 13908 \, 65400 \, 30048 \, 07291 \, 64958 \, 29772 \, 97828 \, 35273 }} {{eqn | o =        | r = 02603 \, 73947 \, 05739 \, 46904 \, 93564 \, 50661 \, 00172 \, 36892 \, 20285 \, 60354 }} {{eqn | o =        | r = 58830 \, 25332 \, 20848 \, 80128 \, 32451 \, 94645 \, 21648 \, 78503 \, 66425 \, 73281 }} {{eqn | o =        | r = 55405 \, 94426 \, 29476 \, 00573 \, 05011 \, 86259 \, 25148 \, 08537 \, 31389 \, 24832 }} {{eqn | o =        | r = 90593 \, 45279 \, 70389 \, 89332 \, 87614 \, 90279 \, 77417 \, 70009 \, 37843 \, 56718 }} {{eqn | o =        | r = 78965 \, 55090 \, 40413 \, 05491 \, 45610 \, 39734 \, 55313 \, 36378 \, 82326 \, 51747 }} {{eqn | o =        | r = 26323 \, 96872 \, 58800 \, 36097 \, 85595 \, 50576 \, 58179 \, 78961 \, 56439 \, 38001 }} {{eqn | o =        | r = 61356 \, 42993 \, 82918 \, 89157 \, 64818 \, 24068 \, 61810 \, 98754 \, 13407 \, 25598 }} {{eqn | o =        | r = 81076 \, 88939 \, 65566 \, 79970 \, 94454 \, 12508 \, 20606 \, 03037 \, 82723 \, 11003 }} {{eqn | o =        | r = 86445 \, 85147 \, 95431 \, 68421 \, 48123 \, 63910 \, 96321 \, 63833 \, 76594 \, 77873 }} {{eqn | o =        | r = 36044 \, 25100 \, 46756 \, 76942 \, 21197 \, 98655 \, 69863 \, 08993 \, 13991 \, 54810 }} {{eqn | o =        | r = 29955 \, 71299 \, 30916 \, 19908 \, 66968 \, 53268 \, 78801 \, 17165 \, 95377 \, 09390 }} {{eqn | o =        | r = 12417 \, 99779 \, 38952 \, 06419 \, 62790 \, 94932 \, 21996 \, 15477 \, 09894 \, 18755 }} {{eqn | o =        | r = 79741 \, 05192 \, 62661 \, 21081 \, 92384 \, 45257 \, 78675 \, 87928 \, 74768 \, 12218 }} {{eqn | o =        | r = 63148 \, 68786 \, 76854 \, 53862 \, 69957 \, 63612 \, 71978 \, 31119 \, 74476 \, 86496 }} {{eqn | o =        | r = 45065 \, 87748 \, 91053 \, 15072 \, 63384 \, 65410 \, 90174 \, 27502 \, 19115 \, 20006 }} {{eqn | o =        | r = 99485 \, 86281 \, 23536 \, 18641 \, 48374 \, 90557 \, 49920 \, 15285 \, 92211 \, 19416 }} {{eqn | o =        | r = 75209 \, 57766 \, 75409 \, 22211 \, 29543 \, 79999 \, 81129 \, 89523 \, 59262 \, 62800 }} {{eqn | o =        | r = 46942 \, 15484 \, 08243 \, 63610 \, 64351 \, 53563 \, 01617 \, 42451 \, 12051 \, 59183 }} {{eqn | o =        | r = 34354 \, 13049 \, 42449 \, 46301 \, 59875 \, 51181 \, 09280 \, 53716 \, 57952 \, 29658 }} {{eqn | o =        | r = 01206 \, 92006 \, 20396 \, 63689 \, 45859 \, 75910 \, 58626 \, 38955 \, 88424 \, 79023 }} {{eqn | o =        | r = 70325 \, 29477 \, 90965 \, 29020 \, 39505 \, 24422 \, 75678 \, 32327 \, 27410 \, 18290 }} {{eqn | o =        | r = 15226 \, 89958 \, 01677 \, 48481 \, 42430 \, 49977 \, 81717 \, 47239 \, 67104 \, 08734 }} {{eqn | o =        | r = 21063 \, 13953 \, 69197 \, 18416 \, 66197 \, 78782 \, 49199 \, 73757 \, 81152 \, 15777 }} {{eqn | o =        | r = 88246 \, 98396 \, 88365 \, 29090 \, 59197 \, 96301 \, 79613 \, 87838 \, 71578 \, 75079 }} {{eqn | o =        | r = 17192 \, 38121 \, 06694 \, 45136 \, 51899 \, 17332 \, 26537 \, 65466 \, 92624 \, 57805 }} {{eqn | o =        | r = 18650 \, 91862 \, 60159 \, 38818 \, 25424 \, 40894 \, 26520 \, 87364 \, 29048 \, 52293 }} {{eqn | o =        | r = 88924 \, 40043 \, 51 }} {{end-eqn}}


Titanic Prime whose Digits are all 9 except for one 1

The integer defined as: :$2 \times 10^{3020} - 1$ is a titanic prime all of whose digits are $9$ except one, which is $1$. That is: :$1 \left({9}\right)_{3020}$ where $\left({a}\right)_b$ means $b$ instances of $a$ in a string.


Titanic Prime consisting of Blocks of 111 of each Digit plus Zeroes

The integer defined as: :$\left({1}\right)_{111} \left({2}\right)_{111} \left({3}\right)_{111} \left({4}\right)_{111} \left({5}\right)_{111} \left({6}\right)_{111} \left({7}\right)_{111} \left({8}\right)_{111} \left({9}\right)_{111} \left({0}\right)_{2284} 1$ where $\left({a}\right)_b$ means $b$ instances of $a$ in a string, is a titanic prime.


Pair of Titanic Twin Primes

The integers defined as: :$190 \, 116 \times 3003 \times 10^{5120} \pm 1$ are a pair of titanic twin primes. That is: :$570 \, 918 \, 347 \paren 9_{5820}$ and: :$570 \, 918 \, 348 \paren 0_{5819} 1$ where $\paren a_b$ means $b$ instances of $a$ in a string.


Indexed Iterated Operation does not Change under Permutation

Let $G$ be a commutative semigroup. Let $a, b \in\Z$ be integers. Let $\closedint a b$ be the integer interval between $a$ and $b$. Let $f: \closedint a b \to G$ be a mapping. Let $\sigma: \closedint a b \to \closedint a b$ be a permutation.


Summation over Cartesian Product as Double Summation

Let $\mathbb A$ be one of the standard number systems $\N, \Z, \Q, \R, \C$. Let $S, T$ be finite sets. Let $S \times T$ be their cartesian product. Then we have an equality of summations over finite sets: :$\displaystyle \sum_{s \mathop \in S} \sum_{t \mathop \in T} \map f {s, t} = \sum_{\paren {s, t} \mathop \in S \times T} \map f {s, t}$


Zeroth Hyperoperation is Successor Function

The '''zeroth hyperoperation''' is the successor function: :$H_0 \left({x, y}\right) = y + 1$


First Hyperoperation is Addition Operation

The '''$1$st hyperoperation''' is the addition operation restricted to the positive integers: :$\forall x, y \in \Z_{\ge 0}: H_1 \left({x, y}\right) = x + y$


Second Hyperoperation is Multiplication Operation

The '''$2$nd hyperoperation''' is the multiplication operation restricted to the positive integers: :$\forall x, y \in \Z_{\ge 0}: H_2 \left({x, y}\right) = x \times y$


Third Hyperoperation is Integer Power Operation

The '''$3$rd hyperoperation''' is the integer power operation restricted to the positive integers: :$\forall x, y \in \Z_{\ge 0}: H_3 \left({x, y}\right) = x^y$


Fourth Hyperoperation is Tetration Operation

The '''$4$th hyperoperation''' is the tetration operation restricted to the positive integers: :$\forall x, y \in \Z_{\ge 0}: H_4 \left({x, y}\right) = x \uparrow \uparrow y$ where $\uparrow \uparrow$ denotes tetration: :$x \uparrow \uparrow n := \begin{cases} 1 & : n = 0 \\ x \uparrow \left({x \uparrow \uparrow \left({n - 1}\right)}\right) & : n > 0 \\ \end{cases}$ In this context, $x \uparrow y$ is the Knuth notation for powers: :$x \uparrow y := x^y$


Largest Integer Expressible by 3 Digits

The largest integer that can be represented using no more than $3$ digits, with no additional symbols, is: :$9^{9^9} = 9^{387 \, 420 \, 489}$ and (at $369 \, 693 \, 100$ digits, is too large to be calculated on a conventional calculator. Note that this does not include the notation for tetration: ${}^9 9$.


Largest Integer Expressible by 3 Digits/Logarithm Base 10

:$\map {\log_{10} } {9^{9^9} } \approx 369 \, 693 \,099 \cdotp 63157 \, 03685 \, 87876 \, 1$


Largest Integer Expressible by 3 Digits/Number of Digits

:$9^{9^9}$ has $369 \, 693 \, 100$ digits when expressed in decimal notation.


Number of Primes up to n Approximates to Eulerian Logarithmic Integral

The prime-counting function approximates to the Eulerian logarithmic integral: :$\map \pi n \approx \displaystyle \int_2^n \frac {\d x} {\ln x}$


Point at which Prime-Counting Function becomes less than Eulerian Logarithmic Integral

Let $\map \pi n$ denote the prime-counting function. Let $a \uparrow b$ be interpreted as Knuth notation for $a^b$.


Reciprocals of Prime Numbers

The decimal representations of the reciprocals of the first few prime numbers are as follows: :{| border="1" |- ! align="right" style = "padding: 2px 10px" | $n$  ! align="left" style = "padding: 2px 10px" | $1 / n$  ! style = "padding: 2px 10px" | Also see |- | align="right" style = "padding: 2px 10px" | $2$ | align="left" style = "padding: 2px 10px" | $0 \cdotp 5$ | style = "padding: 2px 10px" |  |- | align="right" style = "padding: 2px 10px" | $3$ | align="left" style = "padding: 2px 10px" | $0 \cdotp \dot 3$ | style = "padding: 2px 10px" |  |- |- | align="right" style = "padding: 2px 10px" | $5$ | align="left" style = "padding: 2px 10px" | $0 \cdotp 2$ | style = "padding: 2px 10px" |  |- |- | align="right" style = "padding: 2px 10px" | $7$ | align="left" style = "padding: 2px 10px" | $0 \cdotp \dot 14285 \, \dot 7$ | style = "padding: 2px 10px" | Period of Reciprocal of 7 is of Maximal Length |- | align="right" style = "padding: 2px 10px" | $11$ | align="left" style = "padding: 2px 10px" | $0 \cdotp \dot 0 \dot 9$ | style = "padding: 2px 10px" |  |- | align="right" style = "padding: 2px 10px" | $13$ | align="left" style = "padding: 2px 10px" | $0 \cdotp \dot 07692 \dot 3$ | style = "padding: 2px 10px" |  |- | align="right" style = "padding: 2px 10px" | $17$ | align="left" style = "padding: 2px 10px" | $0 \cdotp \dot 05882 \, 35294 \, 11764 \, \dot 7$ | style = "padding: 2px 10px" | Period of Reciprocal of 17 is of Maximal Length |- | align="right" style = "padding: 2px 10px" | $19$ | align="left" style = "padding: 2px 10px" | $0 \cdotp \dot 05263 \, 15789 \, 47368 \, 42 \dot 1$ | style = "padding: 2px 10px" | Period of Reciprocal of 19 is of Maximal Length |- | align="right" style = "padding: 2px 10px" | $23$ | align="left" style = "padding: 2px 10px" | $0 \cdotp \dot 04347 \, 82608 \, 69565 \, 21739 \, 1 \dot 3$ | style = "padding: 2px 10px" | Period of Reciprocal of 23 is of Maximal Length |- | align="right" style = "padding: 2px 10px" | $29$ | align="left" style = "padding: 2px 10px" | $0 \cdotp \dot 03448 \, 27586 \, 20689 \, 65517 \, 24137 \, 93 \dot 1$ | style = "padding: 2px 10px" |  |- | align="right" style = "padding: 2px 10px" | $31$ | align="left" style = "padding: 2px 10px" | $0 \cdotp \dot 03225 \, 80645 \, 1612 \dot 9$ | style = "padding: 2px 10px" | Period of Reciprocal of 31 is of Odd Length |- | align="right" style = "padding: 2px 10px" | $37$ | align="left" style = "padding: 2px 10px" | $0 \cdotp \dot 02 \dot 7$ | style = "padding: 2px 10px" | Period of Reciprocal of 37 has Length 3 |}


Factors of Repunits

The prime factors of the first few repunits are as follows: :{| border="1" |- ! align="right" style = "padding: 2px 10px" | $R_n$  ! align="left" style = "padding: 2px 10px" | Factors of $n$  ! style = "padding: 2px 10px" | Also see |- | align="right" style = "padding: 2px 10px" | $2$ | align="left" style = "padding: 2px 10px" | $11$ | style = "padding: 2px 10px" |  |- | align="right" style = "padding: 2px 10px" | $3$ | align="left" style = "padding: 2px 10px" | $3 \times 37$ | style = "padding: 2px 10px" |  |- |- | align="right" style = "padding: 2px 10px" | $4$ | align="left" style = "padding: 2px 10px" | $11 \times 101$ | style = "padding: 2px 10px" |  |- |- | align="right" style = "padding: 2px 10px" | $5$ | align="left" style = "padding: 2px 10px" | $41 \times 271$ | style = "padding: 2px 10px" |  |- | align="right" style = "padding: 2px 10px" | $6$ | align="left" style = "padding: 2px 10px" | $3 \times 7 \times 11 \times 13 \times 37$ | style = "padding: 2px 10px" |  |- | align="right" style = "padding: 2px 10px" | $7$ | align="left" style = "padding: 2px 10px" | $239 \times 4649$ | style = "padding: 2px 10px" |  |- | align="right" style = "padding: 2px 10px" | $8$ | align="left" style = "padding: 2px 10px" | $11 \times 73 \times 101 \times 137$ | style = "padding: 2px 10px" |  |- | align="right" style = "padding: 2px 10px" | $9$ | align="left" style = "padding: 2px 10px" | $3^2 \times 37 \times 333 \, 667$ | style = "padding: 2px 10px" |  |- | align="right" style = "padding: 2px 10px" | $10$ | align="left" style = "padding: 2px 10px" | $11 \times 41 \times 271 \times 9091$ | style = "padding: 2px 10px" |  |- | align="right" style = "padding: 2px 10px" | $11$ | align="left" style = "padding: 2px 10px" | $21 \, 649 \times 513 \, 239$ | style = "padding: 2px 10px" |  |- | align="right" style = "padding: 2px 10px" | $12$ | align="left" style = "padding: 2px 10px" | $3 \times 7 \times 11 \times 13 \times 37 \times 101 \times 9901$ | style = "padding: 2px 10px" |  |- | align="right" style = "padding: 2px 10px" | $13$ | align="left" style = "padding: 2px 10px" | $53 \times 79 \times 265 \, 371 \, 653$ | style = "padding: 2px 10px" |  |}


Separability is not Weakly Hereditary

The property of separability is not weakly hereditary.


Filter is Finer iff Sets of Basis are Subsets

Let $S$ be a set. Let $\powerset S$ denote the power set of $S$. Let $\FF, \FF' \subset \powerset S$ be two filters on $S$. Let $\FF$ have a basis $\BB$. Let $\FF'$ have a basis $\BB'$. $\FF$ is finer than $\FF'$ {{iff}} for every set of $\BB'$, there is a set of $\BB$ subset to it.


Existence of Topological Space which satisfies no Separation Axioms but T0

There exists at least one example of a topological space for which none of the Tychonoff separation axioms are satisfied except for the $T_0$ (Kolmogorov) axiom.


Existence of Topological Space which satisfies no Separation Axioms but T0 and T1

There exists at least one example of a topological space for which none of the Tychonoff separation axioms are satisfied except for the $T_0$ (Kolmogorov) axiom and $T_1$ (Fréchet) axiom.


Equivalence of Definitions of Unital Associative Commutative Algebra

Let $A$ be a commutative ring with unity.


Existence of Hausdorff Space which is not T3, T4 or T5

There exists at least one example of a topological space which is a $T_2$ (Hausdorff) space (and hence also $T_0$ (Kolmogorov) and $T_1$ (Fréchet), but is not a $T_3$ space, a $T_4$ space or a $T_5$ space.


Existence of Topological Space which satisfies no Separation Axioms but T3

There exists at least one example of a topological space for which none of the Tychonoff separation axioms are satisfied except for the $T_3$ axiom.


Existence of Topological Space which satisfies no Separation Axioms but T4

There exists at least one example of a topological space for which none of the Tychonoff separation axioms are satisfied except for the $T_4$ axiom.


Existence of Tychonoff Space which is not Normal

There exists at least one example of a topological space which is a Tychonoff space, but is not also a normal space.


Existence of T4 Space which is not T3 1/2

There exists at least one example of a topological space which is a $T_4$ space, but is not also a $T_{3 \frac 1 2}$ space.


Existence of Normal Space which is not Completely Normal

There exists at least one example of a topological space which is a normal space, but is not also a completely normal space.


Separation Properties Not Preserved by Expansion

These separation properties are not generally preserved under expansion: :$T_3$ Space :Regular Space :$T_4$ Space :Completely Regular Space :$T_5$ Space :Normal Space :Completely Normal Space


Existence of Completely Normal Space whose Product Space is Not Normal

There exists at least one example of a completely normal topological space $T$ such that the product space $T \times T$ is not a normal topological space.


Existence of Completely Normal Space which is not Perfectly Normal

There exists at least one example of a completely normal topological space which is not perfectly normal.


Existence of Semiregular Topological Space which is not T3

There exists at least one example of a semiregular topological space which is not a $T_3$ space.


Existence of Semiregular Topological Space which is not Completely Hausdorff

There exists at least one example of a semiregular topological space which is not a completely Hausdorff space.


Existence of Sigma-Compact Space which is not Compact

There exists at least one example of a $\sigma$-compact topological space which is not also a compact space.


Existence of Lindelöf Space which is not Sigma-Compact


Existence of Countably Compact Space which is not Sequentially Compact

There exists at least one example of a countably compact topological space which is not also a sequentially compact space.


Existence of Weakly Countably Compact Space which is not Countably Compact

There exists at least one example of a weakly countably compact topological space which is not also a countably compact space.


Existence of Pseudocompact Space which is not Countably Compact

There exists at least one example of a pseudocompact topological space which is not also a countably compact space.


Vectorization of Product of Three Matrices

Let $R$ be a ring. Let $A, B, C$ be matrices over $R$ such that the matrix product $ABC$ is defined. Then $\map {\operatorname {vec} }{ABC} = \paren {C^\intercal \otimes A} \cdot \map {\operatorname {vec} } B$ where: :$\operatorname {vec}$ denotes vectorization :$C^\intercal$ is the transpose of $C$ :$\otimes$ denotes Kronecker product :$\cdot$ denotes matrix product


Existence of Weakly Sigma-Locally Compact Space which is not Strongly Locally Compact

There exists at least one example of a weakly $\sigma$-locally compact topological space which is not also a strongly locally compact space.


Existence of First-Countable Space which is not Second-Countable

There exists at least one example of a first-countable topological space which is not also a second-countable space.


Existence of Lindelöf Space which is not Second-Countable

There exists at least one example of a Lindelöf space which is not also a second-countable space.


Existence of Topological Space satisfying Countable Chain Condition which is not Separable

There exists at least one example of a topological space which satisfies the countable chain condition which is not also a separable space.


Equivalence of Definitions of Strongly Locally Compact Space

Let $T = \left({S, \tau}\right)$ be a topological space. {{TFAE|def = Strongly Locally Compact Space}}


Uniqueness of Representing Objects

Let $C$ be a locally small category. Let $\mathbf{Set}$ be the category of sets. Let $F : \mathbf C \to \mathbf{Set}$ be a covariant functor. Let $(A, \eta)$ and $(B, \xi)$ be representations of $F$. Then there exists a unique isomorphism $f : A \to B$ such that $\eta \circ h^f = \xi$, where: :$h^f$ is the precomposition natural transformation :$\circ$ denotes vertical composition of natural transformations


Existence and Uniqueness of Reduced Form of Group Word

Let $X$ be a set. Let $w$ be a group word on $X$. Then $w$ has a unique reduced form.


Existence of Metacompact Space which is not Paracompact

There exists at least one example of a metacompact topological space which is not also a paracompact space.


Existence of Minimal Hausdorff Space which is not Compact

Let $S$ be a set. Let $\tau$ be the minimal subset of the power set $\powerset S$ such that $\struct {S, \tau}$ is a Hausdorff space. Then it is not necessarily the case that $\struct {S, \tau}$ is compact.


Existence of Maximal Compact Topological Space which is not Hausdorff

Let $S$ be a set. Let $\tau$ be the minimal subset of the power set $\powerset S$ such that $\struct {S, \tau}$ is a compact topological space. Then it is not necessarily the case that $\struct {S, \tau}$ is a Hausdorff space.


Gauss's Lemma (Polynomial Theory)

Gauss's lemma on polynomials may refer to any of the following statements.


Infinite Product of Sigma-Compact Spaces is not always Sigma-Compact

Let $I$ be an indexing set with infinite cardinality. Let $\family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be a family of topological spaces indexed by $I$. Let $\displaystyle \struct {S, \tau} = \prod_{\alpha \mathop \in I} \struct {S_\alpha, \tau_\alpha}$ be the product space of $\family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$. Let each of $\struct {S_\alpha, \tau_\alpha}$ be $\sigma$-compact. Then it is not necessarily the case that $\struct {S, \tau}$ is also $\sigma$-compact.


Uncountable Product of Sequentially Compact Spaces is not always Sequentially Compact

Let $I$ be an indexing set with uncountable cardinality. Let $\family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be a family of topological spaces indexed by $I$. Let $\displaystyle \struct {S, \tau} = \prod_{\alpha \mathop \in I} \struct {S_\alpha, \tau_\alpha}$ be the product space of $\family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$. Let each of $\struct {S_\alpha, \tau_\alpha}$ be sequentially compact. Then it is not necessarily the case that $\struct {S, \tau}$ is also sequentially compact.


Product of Countably Compact Spaces is not always Countably Compact

Let $I$ be an indexing set. Let $\family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be a family of topological spaces indexed by $I$. Let $\displaystyle \struct {S, \tau} = \prod_{\alpha \mathop \in I} \struct {S_\alpha, \tau_\alpha}$ be the product space of $\family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$. Let each of $\struct {S_\alpha, \tau_\alpha}$ be countably compact. Then it is not necessarily the case that $\struct {S, \tau}$ is also countably compact.


Product of Lindelöf Spaces is not always Lindelöf

Let $I$ be an indexing set. Let $\family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be a family of topological spaces indexed by $I$. Let $\displaystyle \struct {S, \tau} = \prod_{\alpha \mathop \in I} \struct {S_\alpha, \tau_\alpha}$ be the product space of $\family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$. Let each of $\struct {S_\alpha, \tau_\alpha}$ be a Lindelöf space. Then it is not necessarily the case that $\struct {S, \tau}$ is also Lindelöf space.


Uncountable Product of Second-Countable Spaces is not always Second-Countable

Let $I$ be an indexing set with uncountable cardinality. Let $\family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be a family of topological spaces indexed by $I$. Let $\displaystyle \struct {S, \tau} = \prod_{\alpha \mathop \in I} \struct {S_\alpha, \tau_\alpha}$ be the product space of $\family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$. Let each of $\struct {S_\alpha, \tau_\alpha}$ be a second-countable space. Then it is not necessarily the case that $\struct {S, \tau}$ is also a second-countable space.


Paracompactness is not always Preserved under Open Continuous Mapping

Let $T_A = \struct {X_A, \tau_A}$ be a topological space which is paracompact. Let $T_B = \struct {X_B, \tau_B}$ be another topological space. Let $\phi: T_A \to T_B$ be a mapping which is both continuous and open. Then it is not necessarily the case that $T_B$ is also paracompact.


Correspondence Theorem for Ring Epimorphisms

Let $A$ and $B$ be commutative rings with unity. Let $\pi : A \to B$ be a ring epimorphism. Let $I$ be the set of ideals of $A$ containing the kernel $\operatorname{ker} \pi$. Let $J$ be the set of ideals of $B$.


Union of Closure with Closure of Complement is Whole Space

Let $T = \left({S, \tau}\right)$ be a topological space. Let $H \subseteq S$ be a subset of $S$. Let $H^-$ denote the closure of $H$ in $T$. Let $S \setminus H$ denote the complement of $H$ relative to $S$. Then: :$H^- \cup \left({S \setminus H}\right)^- = S$


Equivalent Characterisations of Irrational Periodic Continued Fraction

Let $x \in \R \setminus \Q$ be an irrational number. Let $(a_n)_{n\geq 0}$ be its continued fraction. Let $N \geq 0$ be a natural number. {{TFAE}} :$(1):\quad$ The sequence of partial quotients $(a_n)_{n\geq 0}$ is periodic for $n \geq N$. :$(2):\quad$ The sequence of complete quotients $(x_n)_{n\geq 0}$ is periodic for $n \geq N$. :$(3):\quad$ There exists $M > N$ such that $x_M = x_N$.


Component of Point is not always Intersection of its Clopen Sets

Let $T = \struct {S, \tau}$ be a topological space. Let $x \in S$. Let $\map {\operatorname {Comp}_x} T$ denote the component of $x$ in $T$. Let $K_x = \displaystyle \bigcap_{x \mathop \in K} K$ clopen in $T$. Then it is not always the case that $\map {\operatorname {Comp}_x} T = K_x$


Complement of Clopen Set is Clopen

Let $T = \struct {S, \tau}$ be a topological space. Let $H \subseteq S$ be a clopen set of $T$. Let $\relcomp S H$ denote the complement of $H$ relative to $S$. Then $\relcomp S H$ is also a clopen set of $T$.


Clopen Set and Complement form Separation

Let $T = \struct {S, \tau}$ be a topological space. Let $H \subseteq S$ be a clopen set of $T$. Let $\relcomp S H$ be the complement of $H$ relative to $S$. Then $H$ and $\relcomp S H$ form a separation of $T$.


Path Component is not necessarily Arc Component

Let $T = \struct {S, \tau}$ be a topological space. Let $P$ be a path component of $T$. Then it is not necessarily the case that $P$ is also an arc component of $T$.


Component is not necessarily Path Component

Let $T = \struct {S, \tau}$ be a topological space. Let $C$ be a component of $T$. Then it is not necessarily the case that $C$ is also an path component of $T$.


Quasicomponent is not necessarily Component

Let $T = \struct {S, \tau}$ be a topological space. Let $Q$ be a quasicomponent of $T$. Then it is not necessarily the case that $C$ is also a component of $T$.


Simple Infinite Continued Fraction is Uniquely Determined by Limit

Let $(a_n)_{n\geq 0}$ and $(b_n)_{n\geq 0}$ be simple infinite continued fractions in $\R$. Let $(a_n)_{n\geq 0}$ and $(b_n)_{n\geq 0}$ have the same limit. Then they are equal.


Continued Fraction Expansion of Irrational Number Converges to Number Itself

Let $x$ be an irrational number. Then the continued fraction expansion of $x$ converges to $x$.


Continued Fraction Expansion of Limit of Simple Infinite Continued Fraction equals Expansion Itself

Let Let $(a_n)_{n\geq 0}$ be a simple infinite continued fractions in $\R$. Then $(a_n)_{n\geq 0}$ converges to an irrational number, whose continued fraction expansion is $(a_n)_{n\geq 0}$.


Correspondence between Irrational Numbers and Simple Infinite Continued Fractions

Let $\R \setminus \Q$ be the set of irrational numbers. Let $S$ be the set of all simple infinite continued fractions in $\R$. The mappings: :$\R \setminus \Q \to S$ that sends an irrational number to its continued fraction expansion :$S \to \R \setminus \Q$ that sends a simple infinite continued fractions to its value are inverses of each other.


Ultraconnected Space is Connected

Let $T = \struct {S, \tau}$ be a topological space which is ultraconnected. Then $T$ is connected.


Irreducible Space is not necessarily Path-Connected

Let $T = \left({S, \tau}\right)$ be a topological space which is irreducible. Then $T$ is not necessarily path-connected.


Ultraconnected Space is not necessarily Arc-Connected

Let $T = \struct {S, \tau}$ be a topological space which is ultraconnected. Then $T$ is not necessarily arc-connected.


Correspondence between Rational Numbers and Simple Finite Continued Fractions

Let $\Q$ be the set of rational numbers. Let $S$ be the set of all simple finite continued fractions in $\Q$, whose last partial quotient is not $1$. The mappings: :$\Q \to S$ that sends an rational number to its continued fraction expansion :$S \to \Q$ that sends a simple finite continued fractions to its value are inverses of each other.


Existence of Non-Locally Connected Space where Components and Quasicomponents are Equal

There exists at least one example of a topological space which is not locally connected, but whose components and quasicomponents are equal.


Locally Connected Space is not necessarily Connected

Let $T = \struct {S, \tau}$ be a topological space which is locally connected. Then it is not necessarily the case that $T$ is also a connected space.


Irreducible Space with Finitely Many Open Sets is Path-Connected

Let $T = \left({S, \tau}\right)$ be a irreducible topological space. Let its topology $\tau$ be finite. Then $T$ is path-connected.


Topological Space with Generic Point is Path-Connected

Let $T = \struct {S, \tau}$ be a topological space. Let $T$ have a generic point $g \in S$. Then $T$ is path-connected.


Floor of Simple Finite Continued Fraction

Let $\sequence {a_k}_{k \mathop \ge 0}$ be a simple finite continued fraction of length $n \ge 0$. Let $x = [a_0, \ldots, a_n]$ be its value. Then the floor of $x$ is the partial denominator $a_0$: :$\floor x = a_0$ unless $n = 1$ and $a_1 = 1$, in which case $x = \floor x = a_0 + 1$.


Value of Finite Continued Fraction of Strictly Positive Real Numbers is Strictly Positive

Let $(a_0, \ldots, a_n)$ be a finite continued fraction in $\R$ of length $n \geq 0$. Let all partial quotients $a_k>0$ be strictly positive. Let $x = [a_0, a_1, \ldots, a_n]$ be its value. Then $x>0$.


Value of Finite Continued Fraction of Real Numbers is at Least First Term

Let $(a_0, \ldots, a_n)$ be a finite continued fraction in $\R$ of length $n \geq 0$. Let the partial quotients $a_k>0$ be strictly positive for $k>0$. Let $x = [a_0, a_1, \ldots, a_n]$ be its value. Then $x \geq a_0$, and $x>a_0$ if the length $n\geq 1$.


Finite Simple Continued Fraction has Rational Value

Let $n \ge 0$ be a natural number. Let $\paren {a_0, \ldots, a_n}$ be a simple finite continued fraction of length $n$. Then its value $\sqbrk {a_0, \ldots, a_n}$ is a rational number.


Accuracy of Convergents of Convergent Simple Infinite Continued Fraction

Let $C = (a_0, a_1, \ldots)$ be an infinite simple continued fraction in $\R$. Let $C$ converge to $x \in \R$. For $n\geq0$, let $C_n = p_n/q_n$ be the $n$th convergent of $C$, where $p_n$ and $q_n$ are the $n$th numerator and denominator. Then for all $n\geq 0$: :$\left\vert x - \dfrac {p_n}{q_n}\right\vert < \dfrac 1{q_nq_{n+1}}$.


Locally Path-Connected Space is not necessarily Locally Arc-Connected

Let $T = \struct {S, \tau}$ be a topological space which is locally path-connected. Then it is not necessarily the case that $T$ is also a locally arc-connected space.


Path-Connected Space is not necessarily Locally Path-Connected

Let $T = \struct {S, \tau}$ be a topological space which is path-connected. Then it is not necessarily the case that $T$ is also locally path-connected.


Lower Bounds for Denominators of Simple Continued Fraction

Let $n \in \N \cup \{\infty\}$ be an extended natural number. Let $\left[{a_0, a_1, a_2, \ldots}\right]$ be a simple continued fraction in $\R$ of length $N$. Let $q_0, q_1, q_2, \ldots$ be its denominators.


Equality of Rational Numbers

Let $a, b, c, d$ be integers, with $b$ and $d$ nonzero. {{TFAE}} :$(1): \quad$ The rational numbers $\dfrac a b$ and $\dfrac c d$ are equal. :$(2): \quad$ The integers $a d$ and $b c$ are equal.


Locally Path-Connected Space is not necessarily Path-Connected

Let $T = \struct {S, \tau}$ be a topological space which is locally path-connected. Then it is not necessarily the case that $T$ is also path-connected.


Arc-Connected Space is not necessarily Locally Arc-Connected

Let $T = \struct {S, \tau}$ be a topological space which is arc-connected. Then it is not necessarily the case that $T$ is also locally arc-connected.


Locally Arc-Connected Space is not necessarily Arc-Connected

Let $T = \struct {S, \tau}$ be a topological space which is locally arc-connected. Then it is not necessarily the case that $T$ is also arc-connected.


Odd Convergents of Simple Continued Fraction are Strictly Decreasing

The odd convergents satisfy $C_1 > C_3 > C_5 > \cdots$


Even Convergents of Simple Continued Fraction are Strictly Increasing

The even convergents satisfy $C_0 < C_2 < C_4 \cdots$.


Denominators of Simple Continued Fraction are Strictly Positive

Let $n \in \N \cup \{\infty\}$ be an extended natural number. Let $(a_0, a_1, \ldots)$ be a simple continued fraction in $\R$ of length $n$. Let $q_0, q_1, q_2, \ldots$ be its denominators. Then for $0 \leq k \leq n$ we have $q_k > 0$.


Simple Finite Continued Fraction is Almost Determined by Value

Let $n,m \geq 0$ be natural number. Let $(a_k)_{0 \leq k \leq m}$ and $(b_k)_{0 \leq k \leq n}$ be simple finite continued fractions in $\R$. Let $(a_k)_{0 \leq k \leq m}$ and $(b_k)_{0 \leq k \leq n}$ have the same value. Then either: *$n = m$, and the sequences are equal. *$n = m + 1$, $a_k = b_k$ for $k \leq m$, $a_m = b_m-1$ and $b_{m+1} = 1$ *$m = n + 1$, $a_k = b_k$ for $k \leq n$, $b_n = a_n-1$ and $a_{n+1} = 1$


Subgroup Generated by One Element is Set of Powers

Let $G$ be a group. Let $a \in G$. Then the subgroup generated by $a$ is the set of powers: :$\gen a  = \set {a^n : n \in \Z}$


Finite Product Space is Connected iff Factors are Connected

Let $T_1 = \struct {S_1, \tau_1}, T_2 = \struct {S_2, \tau_2}, \dotsc, T_n = \struct {S_n, \tau_n}$ be topological spaces. Let $T = \displaystyle \prod_{i \mathop = 1}^n T_i$ be the product space of $T_1, T_2, \ldots, T_n$. Then $T$ is connected {{iff}} each of $T_1, T_2, \ldots, T_n$ are connected.


Binary Coproduct in Preadditive Category is Biproduct

Let $A$ be a preadditive category. Let $a_1, a_2$ be objects of $A$. Let $(a_1 \sqcup a_2, i_1, i_2)$ be their binary coproduct, assuming it exists. Let $p_1 : a_1 \sqcup a_2 \to a_1$ be the unique morphism with: :$p_1 \circ i_1 = 1 : a_1 \to a_1$ :$p_1 \circ i_2 = 0 : a_1 \to a_2$ Let $p_2 : a_1 \sqcup a_2 \to a_2$ be the unique morphism with: :$p_2 \circ i_1 = 0 : a_2 \to a_1$ :$p_2 \circ i_2 = 1 : a_2 \to a_2$ where $1$ is the identity morphism and $0$ is the zero morphism. Then $(a_1 \sqcup a_2, i_1, i_2, p_1, p_2)$ is the binary biproduct of $a_1$ and $a_2$.


Continuous Mapping from Compact Space to Hausdorff Space Preserves Local Connectedness

Let $T_1 = \struct {S_1, \tau_1}$ be a compact topological space. Let $T_2 = \struct {S_2, \tau_2}$ be a $T_2$ (Hausdorff) space. Let $f: T_1 \to T_2$ be a continuous mapping. Let $T_1$ be locally connected. Then $T_2$ is also locally connected.


Minimum Degree Bound for Simple Planar Graph

Let $G$ be a simple connected planar graph. Then:  :$\map \delta G \le 5$ where $\map \delta G$ denotes the minimum degree of vertices of $G$.


Linear Bound Lemma

For a simple connected planar graph $G_n$, where $n \geq 3$ is a number of vertices: :$m \leq 3 n − 6$, where $m$ is a number of edges.


Existence of Connected Space which is Totally Pathwise Disconnected

There exists at least one example of a topological space which is both connected and totally pathwise disconnected.


Zero Dimensional Space is not necessarily T0

Let $T = \struct {S, \tau}$ be a zero dimensional topological space. Then $T$ is not necessarily a $T_0$ (Kolmogorov) space.


Scattered Space is not necessarily T1

Let $T = \struct {S, \tau}$ be a scattered topological space. Then $T$ is not necessarily a $T_1$ (Fréchet) space.


Existence of Connected Non-T1 Scattered Space

There exists at least one example of a connected topological space which is not a $T_1$ (Fréchet) space, which is also a scattered space.


Sorgenfrey Line is not Second-Countable

Let $T = \struct {\mathbb R, \tau}$ be the Sorgenfrey line. Then $T$ is not second-countable.


Existence of Biconnected Set without Dispersion Point

There exists at least one example of a biconnected set which does not have a dispersion point.


Existence of Connected Punctiform Space

There exists at least one example of a connected topological space which is also punctiform.


Metric Space is Completely Normal

Let $M = \struct {A, d}$ be a metric space. Then $M$ is a completely normal space.


Bounded Metric Space is not necessarily Totally Bounded

Let $M = \struct {A, d}$ be a bounded metric space. Then it is not necessarily the case that $M$ is totally bounded.


Total Boundedness is not Preserved under Homeomorphism

Let $M = \struct {A, d}$ be a totally bounded metric space. Let $M' = \struct {A', d'}$ be a metric space. Let $M$ be homeomorphic to $M'$. Then it is not necessarily the case that $M'$ is totally bounded.


Topological Completeness is not Hereditary

Let $T = \struct {S, \tau}$ be a topological space which is topologically complete. Let $H \subseteq S$ be a subset of $S$. Let $\struct {H, \tau_H}$ be the topological subspace of $T$ induced by $H$. Then it is not necessarily the case that $\struct {H, \tau_H}$ is also topologically complete. That is, topological completeness is not hereditary.


Completion Theorem (Metric Space)/Lemma 4

:$\tilde M = \struct {\tilde A, \tilde d}$ is unique up to isometry.


Sorgenfrey Line is First-Countable

Let $\R$ be the set of real numbers. Let $\BB = \set {\hointr a b: a, b \in \R}$. Let $\tau$ be the topology generated by $\BB$, that is, the Sorgenfrey line. Then $\tau$ is first-countable.


Metrizable Space is not necessarily Second-Countable

Let $T = \struct {S, \tau}$ be a topological space which is metrizable. Then it is not necessarily the case that $T$ is second-countable.


Regular Paracompact Space is not necessarily Metrizable

Let $T = \struct {S, \tau}$ be a topological space which is regular and paracompact. Then it is not necessarily the case that $T$ is metrizable.


Uniform Space whose Topology is Metrizable is not necessarily Metrizable

Let $\UU$ be a uniformity on a set $S$. Let $\struct {\struct {S, \UU}, \tau}$ be the uniform space generated from $\UU$. Let $T = \struct {S, \tau}$ be the uniformizable space yielded by $\struct {\struct {S, \UU}, \tau}$. Let $T$ be a metrizable space. Then it is not necessarily the case that $\UU$ is itself a metrizable uniformity.


Open Sets in Indiscrete Topology

$H$ is an open set of $T$ {{iff}} either $H = S$ or $H = \O$.


Closed Sets in Indiscrete Topology

$H$ is a closed set of $T$ {{iff}} either $H = S$ or $H = \O$.


F-Sigma Sets in Indiscrete Topology

$H$ is an $F_\sigma$ ($F$-sigma) set of $T$ {{iff}} either $H = S$ or $H = \O$.


Subset of Indiscrete Space is Compact

$H$ is compact in $T$.


Partition Topology is not Completely Hausdorff

Let $S$ be a set and let $\PP$ be a partition on $S$ which is not the (trivial) partition of singletons. Let $T = \struct {S, \tau}$ be the partition space whose basis is $\PP$. Then $T$ is not a $T_{2 \frac 1 2}$ (completely Hausdorff) space.


Partition Topology is not Hausdorff

Let $S$ be a set and let $\PP$ be a partition on $S$ which is not the (trivial) partition of singletons. Let $T = \struct {S, \tau}$ be the partition space whose basis is $\PP$. Then $T$ is not a $T_2$ (Hausdorff) space.


Partition Topology is not T1

Let $S$ be a set and let $\PP$ be a partition on $S$ which is not the (trivial) partition of singletons. Let $T = \struct {S, \tau}$ be the partition space whose basis is $\PP$. Then $T$ is not a $T_1$ (Fréchet) space.


Odd-Even Topology is Lindelöf

$T$ is Lindelöf.


Odd-Even Topology is Separable

$T$ is separable.


Odd-Even Topology is First-Countable

$T$ is first-countable.


Particular Point Space less Particular Point is Discrete

Let $T = \left({S, \tau_p}\right)$ be a particular point space, whose particular point is $p$. Let $H = S \setminus \left\{ {p}\right\}$ where $\setminus$ denotes set difference. Then the topological subspace $T_H = \left({H, \tau_H}\right)$ induced on $H$ by $\tau_p$ is a discrete space.


Zero is Accumulation Point of Sequence in Sierpiński Space

Let $T = \struct {\set {0, 1}, \tau_0}$ be a Sierpiński space. The sequence in $T$: :$\sigma = \sequence {0, 1, 0, 1, \ldots}$ has $0$ as an accumulation point.


Infinite Particular Point Space is not Countably Paracompact

Let $T = \left({S, \tau_p}\right)$ be an infinite particular point space. Then $T$ is not countably paracompact.


Closed Extension Topology is not Hausdorff

Let $T = \struct {S, \tau}$ be a topological space. Let $T^*_p = \struct {S^*_p, \tau^*_p}$ be the closed extension space of $T$. Then $T^*_p$ is not a $T_2$ (Hausdorf) space.


Closed Extension Topology is not T3

Let $T = \struct {S, \tau}$ be a topological space. Let $T^*_p = \struct {S^*_p, \tau^*_p}$ be the closed extension space of $T$. Then $T^*_p$ is not a $T_3$ space.


Either-Or Topology is T4

Let $T = \struct {S, \tau}$ be the either-or space. Then $T$ is a $T_4$ space.


Limit Points in Fort Space

Let $T = \struct {S, \tau_p}$ be a Fort space. Let $x \in S$ such that $x \ne p$. Then $p$ is the only limit point of $x$.


Rational Numbers in Real Euclidean Plus Space are Open Set

Let $\R$ be the set of real numbers. Let $d: \R \times \R \to \R$ be the Euclidean plus metric: :$\map d {x, y} := \size {x - y} + \displaystyle \sum_{i \mathop = 1}^\infty 2^{-i} \map \inf {1, \size {\max_{j \mathop \le i} \frac 1 {\size {x - r_j} } - \max_{j \mathop \le i} \frac 1 {\size {y - r_j} } } }$ Let $\Q$ be the set of rational numbers. Then $\Q$ is an open set of $\struct {\R, d}$.


Alexandroff Extension is Compact

Let $T = \struct {S, \tau}$ be a non-empty topological space. Let $p$ be a new element not in $S$. Let $S^* := S \cup \set p$. Let $T^* = \struct {S^*, \tau^*}$ be the Alexandroff extension on $S$. Then $T^*$ is a compact topological space.


Condition for Alexandroff Extension to be T1 Space

Let $T = \struct {S, \tau}$ be a non-empty topological space. Let $p$ be a new element not in $S$. Let $S^* := S \cup \set p$. Let $T^* = \struct {S^*, \tau^*}$ be the Alexandroff extension on $S$. Then $T^*$ is a $T_1$ (Fréchet) space {{iff}} $T$ is a $T_1$ (Fréchet) space.


Condition for Alexandroff Extension to be T2 Space

Let $T = \struct {S, \tau}$ be a non-empty topological space. Let $p$ be a new element not in $S$. Let $S^* := S \cup \set p$. Let $T^* =\struct {S^*, \tau^*}$ be the Alexandroff extension on $S$. Then $T^*$ is a $T_2$ (Hausdorff) space {{iff}} $T$ is a locally compact Hausdorff space.


Alexandroff Extension which is T2 Space is also T4 Space

Let $T = \struct {S, \tau}$ be a non-empty topological space. Let $p$ be a new element not in $S$. Let $S^* := S \cup \set p$. Let $T^* = \struct {S^*, \tau^*}$ be the Alexandroff extension on $S$. Let $T^*$ be a $T_2$ (Hausdorff) space. Then $T^*$ is a $T_4$ space.


Alexandroff Extension of Rational Number Space is not Hausdorff

Let $\struct {\Q, \tau_d}$ be the rational number space under the Euclidean topology $\tau_d$. Let $p$ be a new element not in $\Q$. Let $\Q^* := \Q \cup \set p$. Let $T^* = \struct {\Q^*, \tau^*}$ be the Alexandroff extension on $\struct {\Q, \tau_d}$. Then $T^*$ is not a $T_2$ (Hausdorff) space.


Alexandroff Extension of Rational Number Space is T1 Space

Let $\struct {\Q, \tau_d}$ be the rational number space under the Euclidean topology $\tau_d$. Let $p$ be a new element not in $\Q$. Let $\Q^* := \Q \cup \set p$. Let $T^* = \struct {\Q^*, \tau^*}$ be the Alexandroff extension on $\struct {\Q, \tau_d}$. Then $T^*$ is a $T_1$ (Fréchet) space.


Particular Point of Alexandroff Extension of Rational Number Space is Dispersion Point

Let $\struct {\Q, \tau_d}$ be the rational number space under the Euclidean topology $\tau_d$. Let $p$ be a new element not in $\Q$. Let $\Q^* := \Q \cup \set p$. Let $T^* = \struct {\Q^*, \tau^*}$ be the Alexandroff extension on $\left({\Q, \tau_d}\right)$. Then $p$ is a dispersion point of $T^*$.


Alexandroff Extension of Rational Number Space is Connected

Let $\struct {\Q, \tau_d}$ be the rational number space under the Euclidean topology $\tau_d$. Let $p$ be a new element not in $\Q$. Let $\Q^* := \Q \cup \set p$. Let $T^* = \struct {\Q^*, \tau^*}$ be the Alexandroff extension on $\struct {\Q, \tau_d}$. Then $T^*$ is a connected space.


Alexandroff Extension of Rational Number Space is Biconnected

Let $\struct {\Q, \tau_d}$ be the rational number space under the Euclidean topology $\tau_d$. Let $p$ be a new element not in $\Q$. Let $\Q^* := \Q \cup \set p$. Let $T^* = \struct {\Q^*, \tau^*}$ be the Alexandroff extension on $\struct {\Q, \tau_d}$. Then $T^*$ is a biconnected space.


Hilbert Sequence Space is Complete Metric Space

Let $A$ be the set of all real sequences $\left\langle{x_i}\right\rangle$ such that the series $\displaystyle \sum_{i \mathop \ge 0} x_i^2$ is convergent. Let $\ell^2 = \struct {A, d_2}$ be the Hilbert sequence space on $\R$. Then $\ell^2$ is a complete metric space.


Hilbert Sequence Space is Separable

Let $A$ be the set of all real sequences $\sequence {x_i}$ such that the series $\displaystyle \sum_{i \mathop \ge 0} x_i^2$ is convergent. Let $\ell^2 = \struct {A, d_2}$ be the Hilbert sequence space on $\R$. Then $\ell^2$ is a separable space.


Hilbert Sequence Space is Second-Countable

Let $\ell^2$ be the Hilbert sequence space on $\R$. Then $\ell^2$ is a second-countable space.


Hilbert Sequence Space is Lindelöf

Let $\ell^2$ be the Hilbert sequence space on $\R$. Then $\ell^2$ is a Lindelöf space.


Compact Subset of Hilbert Sequence Space is Closed

Let $A$ be the set of all real sequences $\sequence {x_i}$ such that the series $\displaystyle \sum_{i \mathop \ge 0} x_i^2$ is convergent. Let $\ell^2 = \struct {A, d_2}$ be the Hilbert sequence space on $\R$. Let $H$ be a compact subset of $\ell^2$ Then $H$ is closed in $\ell^2$.


Subset of Hilbert Sequence Space with Non-Empty Interior is not Compact

Let $A$ be the set of all real sequences $\left\langle{x_i}\right\rangle$ such that the series $\displaystyle \sum_{i \mathop \ge 0} x_i^2$ is convergent. Let $\ell^2 = \struct {A, d_2}$ be the Hilbert sequence space on $\R$. Let $H$ be a subset of $\ell^2$ whose interior  is non-empty. Then $H$ is not compact in $\ell^2$.


Point in Hilbert Sequence Space has no Compact Neighborhood

Let $A$ be the set of all real sequences $\left\langle{x_i}\right\rangle$ such that the series $\displaystyle \sum_{i \mathop \ge 0} x_i^2$ is convergent. Let $\ell^2 = \struct {A, d_2}$ be the Hilbert sequence space on $\R$. Then no point of $\ell^2$ has a compact neighborhood.


Compact Subset of Hilbert Sequence Space is Nowhere Dense

Let $A$ be the set of all real sequences $\sequence {x_i}$ such that the series $\displaystyle \sum_{i \mathop \ge 0} x_i^2$ is convergent. Let $\ell^2 = \struct {A, d_2}$ be the Hilbert sequence space on $\R$. Let $H$ be a compact subset of $\ell^2$. Then $H$ is nowhere dense in $\ell^2$.


Hilbert Sequence Space is Arc-Connected

Let $A$ be the set of all real sequences $\sequence {x_i}$ such that the series $\displaystyle \sum_{i \mathop \ge 0} x_i^2$ is convergent. Let $\ell^2 = \struct {A, d_2}$ be the Hilbert sequence space on $\R$. Then $\ell^2$ is arc-connected.


Hilbert Sequence Space is Homeomorphic to Countable Infinite Product of Real Number Spaces

Let $A$ be the set of all real sequences $\sequence {x_i}$ such that the series $\displaystyle \sum_{i \mathop \ge 0} x_i^2$ is convergent. Let $\ell^2 = \struct {A, d_2}$ be the Hilbert sequence space on $\R$. Let $\struct {\R, \tau_d}$ denote the real number line under the Euclidean topology. Let $\R^\omega = \displaystyle \prod_{i \mathop \in \N} \struct {\R, \tau_d}$ denote the countable-dimensional real Cartesian space under the Tychonoff topology. Then $\ell^2$ is homeomorphic to $\R^\omega$.


Fréchet Space (Functional Analysis) is Metric Space

Let $\struct {\R^\omega, d}$ be the '''Fréchet space on $\R^\omega$'''. Then $\struct {\R^\omega, d}$ is a metric space.


Countable Infinite Product of Real Number Spaces is Homeomorphic to Fréchet Metric Space

Let $\struct {\R, \tau_d}$ denote the real number line under the Euclidean topology. Let $T = \struct {\R^\omega, \tau} = \displaystyle \prod_{i \mathop \in \N} \struct {\R, \tau_d}$ denote the countable-dimensional real Cartesian space under the product topology $\tau$. Let $\struct {\R^\omega, d}$ be the '''Fréchet space on $\R^\omega$''', where: :$\map d {x, y} = \displaystyle \sum_{i \mathop \in \N} \dfrac {2^{-i} \size {x_i - y_i} } {1 + \size {x_i - y_i} }$ Then the topology induced by $d$ is exactly the Tychonoff topology $\tau$.


Separable Metric Space is Homeomorphic to Subspace of Fréchet Metric Space

Let $M = \struct {A, d}$ be a metric space whose induced topology is separable. Then $M$ is homeomorphic to a subspace of the Fréchet space $\struct {\R^\omega, d}$ on the countable-dimensional real Cartesian space $\R^\omega$.


Hilbert Cube is Metric Space

Let $M = \struct {I^\omega, d_2}$ be the Hilbert cube. Then $M$ is a metric space.


Hilbert Cube is Homeomorphic to Countable Infinite Product of Real Number Unit Intervals

Let $M_1 = \struct {I^\omega, d_2}$ be the Hilbert cube: :$M_1 = \displaystyle \prod_{k \mathop \in \N} \closedint 0 {\dfrac 1 k}$ under the same metric as that of the Hilbert sequence space: :$\displaystyle \forall x = \sequence {x_i}, y = \sequence {y_i} \in I^\omega: \map {d_2} {x, y} := \paren {\sum_{k \mathop \ge 0} \paren {x_k - y_k}^2}^{\frac 1 2}$ Let $M_2$ be the metric space defined as: :$M_2 = \displaystyle \prod_{k \mathop \in \N} \closedint 0 1$ under the Tychonoff topology. Then $M_1$ and $M_2$ are homeomorphic.


Hilbert Cube is Completely Normal

Let $M = \struct {I^\omega, d_2}$ be the Hilbert cube. Then $M$ is a completely normal space.


Hilbert Cube is Separable

Let $M = \struct {I^\omega, d_2}$ be the Hilbert cube. Then $M$ is a separable space.


Hilbert Cube is Second-Countable

Let $M = \struct {I^\omega, d_2}$ be the Hilbert cube. Then $M$ is a second-countable space.


Hilbert Cube is Arc-Connected

Let $M = \struct {I^\omega, d_2}$ be the Hilbert cube. Then $M$ is an arc-connected space.


Interval of Totally Ordered Set is Convex

Let $\struct {S, \preccurlyeq}$ be a totally ordered set. Let $I \subseteq S$ be an interval in $S$. Then $I$ is convex.


Convex Set of Ordered Set is not necessarily Interval

Let $\struct {S, \preccurlyeq}$ be an ordered set. Let $C$ be a convex set of $S$. Then it is not necessarily the case that $C$ is an interval of $S$.


Union of Non-Disjoint Convex Sets is Convex Set

Let $\struct {S, \preccurlyeq}$ be an ordered set. Let $\CC$ be a set of convex sets of $S$ such that their intersection is non-empty: :$\displaystyle \bigcap \CC \ne \O$ Then the union $\displaystyle \bigcup \CC$ is also convex.


Subset of Convex Set can be Uniquely Expressed as Partition of Maximal Convex Sets

Let $\struct {S, \preccurlyeq}$ be an ordered set. Then $S$ can be uniquely expressed as the partition whose components are maximally convex sets.


Separated Subsets of Linearly Ordered Space under Order Topology

Let $T = \struct {S, \preceq, \tau}$ be a linearly ordered space. Let $A$ and $B$ be separated sets of $T$. Let $A^*$ and $B^*$ be defined as: :$A^* := \ds \bigcup \set {\closedint a b: a, b \in A, \closedint a b \cap B^- = \O}$ :$B^* := \ds \bigcup \set {\closedint a b: a, b \in B, \closedint a b \cap A^- = \O}$ where $A^-$ and $B^-$ denote the closure of $A$ and $B$ in $T$. Then $A^*$ and $B^*$ are themselves separated sets of $T$.


Generators of Special Linear Group of Order 2 over Integers

Let: :$ S = \begin{pmatrix} 0 & - 1 \\ 1 & 0 \end{pmatrix}$ and: :$T = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$ Then $S$ and $T$ are generators for the special linear group of order $2$ over $\Z$.


Partition of Linearly Ordered Space by Convex Components is Linearly Ordered Set

Let $T = \struct {S, \preceq, \tau}$ be a linearly ordered space. Let $A$ and $B$ be separated sets of $T$. Let $A^*$ and $B^*$ be defined as: :$A^* := \displaystyle \bigcup \set {\closedint a b: a, b \in A, \closedint a b \cap B^- = \O}$ :$B^* := \displaystyle \bigcup \set {\closedint a b: a, b \in B, \closedint a b \cap A^- = \O}$ where $A^-$ and $B^-$ denote the closure of $A$ and $B$ in $T$. Let $A^*$, $B^*$ and $\relcomp S {A^* \cup B^*}$ be expressed as the union of convex components of $S$: :$\displaystyle A^* = \bigcup A_\alpha, \quad B^* = \bigcup B_\beta, \quad \relcomp S {A^* \cup B^*} = \bigcup C_\gamma$ where $\relcomp S X$ denotes the complement of $X$ with respect to $S$. Then the set $M = \set {A_\alpha, B_\beta, C_\gamma}$ inherits a linear ordering from $S$, and so is a linearly ordered set.


Successor Sets of Linearly Ordered Set Induced by Convex Component Partition

Let $T = \struct {S, \preceq, \tau}$ be a linearly ordered space. Let $A$ and $B$ be separated sets of $T$. Let $A^*$ and $B^*$ be defined as: :$A^* := \displaystyle \bigcup \set {\closedint a b: a, b \in A, \closedint a b \cap B^- = \O}$ :$B^* := \displaystyle \bigcup \set {\closedint a b: a, b \in B, \closedint a b \cap A^- = \O}$ where $A^-$ and $B^-$ denote the closure of $A$ and $B$ in $T$. Let $A^*$, $B^*$ and $\relcomp S {A^* \cup B^*}$ be expressed as the union of convex components of $S$: :$\displaystyle A^* = \bigcup A_\alpha, \quad B^* = \bigcup B_\beta, \quad \relcomp S {A^* \cup B^*} = \bigcup C_\gamma$ where $\relcomp S X$ denotes the complement of $X$ with respect to $S$. Let $M$ be the linearly ordered set: :$M = \set {A_\alpha, B_\beta, C_\gamma}$ as defined in Partition of Linearly Ordered Space by Convex Components is Linearly Ordered Set. Then each of the sets $A_\alpha \in M$ has an immediate successor in $M$ if $A_\alpha$ intersects the closure of $S_\alpha$, the set of strict upper bounds for $A_\alpha$. Similarly for $B_\beta$. That immediate successor ${C_\alpha}^+$ to $A_\alpha$ is an element in $\set {C_\gamma}$.


Linearly Ordered Space is T5

Let $T = \struct {S, \preceq, \tau}$ be a linearly ordered space. Then $T$ is a $T_5$ space.


Linearly Ordered Space is T1

Let $T = \struct {S, \preceq, \tau}$ be a linearly ordered space. Then $T$ is a $T_1$ (Fréchet) space.


Linearly Ordered Space is Completely Normal

Let $T = \struct {S, \preceq, \tau}$ be a linearly ordered space. Then $T$ is a completely normal space.


Linearly Ordered Space is Compact iff Complete

Let $T = \struct {S, \preceq, \tau}$ be a linearly ordered space. Then $T$ is a compact space {{iff}} it is complete.


Basis for Open Ordinal Topology

Let $\Gamma$ be a limit ordinal. Let $\hointr 0 \Gamma$ denote the open ordinal space on $\Gamma$. Consider the set $\BB$ of subsets of $\hointr 0 \Gamma$ of the form: :$\openint \alpha {\beta + 1} = \hointl \alpha \beta = \set {x \in \hointr 0 \Gamma: \alpha < x < \beta + 1}$ for $\alpha, \beta \in \hointr 0 \Gamma$. {{explain|It's unclear from the source work used whether this is the basis for the Open Ordinal Topology or Closed Ordinal Topology.}} Then $\BB$ forms a basis for $\hointr 0 \Gamma$.


Omega is Closed in Uncountable Closed Ordinal Space but not G-Delta Set

Let $\Omega$ denote the first uncountable ordinal. Let $\closedint 0 \Omega$ denote the closed ordinal space on $\Omega$. Then $\set \Omega$ is a closed set of $\closedint 0 \Omega$ but not a $G_\delta$ set.


Uncountable Closed Ordinal Space is not First-Countable

Let $\Omega$ denote the first uncountable ordinal. Let $\closedint 0 \Omega$ denote the closed ordinal space on $\Omega$. Then $\closedint 0 \Omega$ is not a first-countable space.


Omega as Limit Point of Intervals of Uncountable Closed Ordinal Space

Let $\Omega$ denote the first uncountable ordinal. Let $\closedint 0 \Omega$ denote the closed ordinal space on $\Omega$. Then $\Omega$ is a limit point of the set $\openint a \Omega$, but not the limit point of any sequence of points in $\openint a \Omega$.


Uncountable Open Ordinal Space is not Separable

Let $\Omega$ denote the first uncountable ordinal. Let $\hointr 0 \Omega$ denote the open ordinal space on $\Omega$. Then $\hointr 0 \Omega$ is not a separable space.


Uncountable Closed Ordinal Space is not Separable

Let $\Omega$ denote the first uncountable ordinal. Let $\closedint 0 \Omega$ denote the closed ordinal space on $\Omega$. Then $\closedint 0 \Omega$ is not a separable space.


Uncountable Open Ordinal Space is First-Countable

Let $\Omega$ denote the first uncountable ordinal. Let $\hointr 0 \Omega$ denote the open ordinal space on $\Omega$. Then $\hointr 0 \Omega$ is a first-countable space.


Ordinal Space is Completely Normal

Let $\Gamma$ denote a limit ordinal. Let $\hointr 0 \Gamma$ denote the open ordinal space on $\Gamma$. Let $\closedint 0 \Gamma$ denote the closed ordinal space on $\Gamma$. Then $\hointr 0 \Gamma$ and $\closedint 0 \Gamma$ are both completely normal.


Uncountable Closed Ordinal Space is not Perfectly Normal

Let $\Omega$ denote the first uncountable ordinal. Let $\closedint 0 \Omega$ denote the closed ordinal space on $\Omega$. Then $\closedint 0 \Omega$ is not a perfectly normal space.


Uncountable Closed Ordinal Space is not Second-Countable

Let $\Omega$ denote the first uncountable ordinal. Let $\closedint 0 \Omega$ denote the closed ordinal space on $\Omega$. Then $\closedint 0 \Omega$ is not a second-countable space.


Uncountable Open Ordinal Space is not Second-Countable

Let $\Omega$ denote the first uncountable ordinal. Let $\hointr 0 \Omega$ denote the open ordinal space on $\Omega$. Then $\hointr 0 \Omega$ is not a second-countable space.


Ring Element is Unit iff Unit in Integral Extension

let $A$ be a commutative ring with unity. Let $a \in A$. Let $B$ be an integral ring extension of $A$. {{TFAE}} :$(1): \quad a$ is a unit of $A$ :$(2): \quad a$ is a unit of $B$


Separable Elements Form Field

Let $E/F$ be an algebraic field extension. Then the subset of separable elements of $E$ form a intermediate field, called the '''relative separable closure'''.


Decomposition of Field Extension as Separable Extension followed by Purely Inseparable

Let $E/F$ be an algebraic field extension. Then the relative separable closure $K=F^{sep}$ in $E$ is the unique intermediate field with the following properties: * $K/F$ is separable. * $E/K$ is purely inseparable.


Subextensions of Separable Field Extension are Separable

Let $E/K/F$ be a tower of fields. Let $E/F$ be separable. Then $E/K$ and $K/F$ are separable.


Finite Orbit under Group of Automorphisms of Field implies Separable over Fixed Field

Let $E$ be a field. Let $G \le \Aut E$ be a subgroup of its automorphism group. Let $F = \map {\operatorname {Fix}_E} G$ be its fixed field. Let $\alpha \in E$ have a finite orbit under $G$. Then $\alpha$ is separable over $F$.


Countable Closed Ordinal Space is Second-Countable

Let $\Omega$ denote the first uncountable ordinal. Let $\Gamma$ be a limit ordinal which strictly precedes $\Omega$. Let $\closedint 0 \Gamma$ denote the closed ordinal space on $\Gamma$. Then $\closedint 0 \Gamma$ is a second-countable space.


Countable Open Ordinal Space is Second-Countable

Let $\Omega$ denote the first uncountable ordinal. Let $\Gamma$ be a limit ordinal which strictly precedes $\Omega$. Let $\hointr 0 \Gamma$ denote the open ordinal space on $\Gamma$. Then $\hointr 0 \Gamma$ is a second-countable space.


Minimal Polynomial of Element with Finite Orbit under Group of Automorphisms over Fixed Field in terms of Orbit

Let $E$ be a field. Let $G \le \Aut E$ be a subgroup of its automorphism group. Let $F = \map {\operatorname {Fix_E} } G$ be its fixed field. Let $\alpha \in E$ have a finite orbit under $G$. Then $\alpha$ is algebraic over $F$ and the product of polynomials :$\map p x = \displaystyle \prod_{\beta \mathop \in \Lambda} \paren {x - \beta}$ is the minimal polynomial of $\alpha$ over $F$.


Countable Closed Ordinal Space is Metrizable

Let $\Omega$ denote the first uncountable ordinal. Let $\Gamma$ be a limit ordinal which strictly precedes $\Omega$. Let $\closedint 0 \Gamma$ denote the closed ordinal space on $\Gamma$. Then $\closedint 0 \Gamma$ is a metrizable space.


Closed Ordinal Space is Complete Order Space

Let $\Gamma$ be a limit ordinal. Let $\closedint 0 \Gamma$ denote the closed ordinal space on $\Gamma$. Then $\closedint 0 \Gamma$ is a complete order space.


Closed Ordinal Space is Compact

Let $\Gamma$ be a limit ordinal. Let $\closedint 0 \Gamma$ denote the closed ordinal space on $\Gamma$. Then $\closedint 0 \Gamma$ is a compact space.


Ordinal Space is Strongly Locally Compact

Let $T$ denote an ordinal space on a limit ordinal $\Gamma$. Then $T$ is a strongly locally compact space.


Open Ordinal Space is not Compact in Closed Ordinal Space

Let $\Gamma$ be a limit ordinal. Let $\hointr 0 \Gamma$ denote the open ordinal space on $\Gamma$. Consider the compact subspace $\hointr 0 \Gamma$. Then $\hointr 0 \Gamma$ is not compact in $\closedint 0 \Gamma$.


Integral Transform is Mapping

Let $\map F p$ be an integral transform: :$\map F p = \displaystyle \int_a^b \map f x \map K {p, x} \rd x$ Let $T$ be the integral operator associated with $\map F p$. Then $T$ is a mapping from the domain of $T$ to its image. That is, for every $\map f x$ there exists a unique $\map F p$.


Integral Operator is Linear

Let $T$ be an integral operator. Let $f$ and $g$ be integrable real functions on a domain appropriate to $T$. Then $T$ is a linear operator: :$\forall \alpha, \beta \in \R: \map T {\alpha f + \beta g} = \alpha \map T f + \beta \map T g$


Inverse Integral Operator is Linear if Unique

Let $T$ be an integral operator. Let $f$ be an integrable real function on a domain appropriate to $T$. Let $F = \map T f$ and $G = \map T g$. Let $T$ have a unique inverse $T^{-1}$. Then $T^{-1}$ is a linear operator: :$\forall p, q \in \R: \map {T^{-1} } {p F + q G} = p \map {T^{-1} } F + q \map {T^{-1} } G$


Trigonometric Series is Convergent if Sum of Absolute Values of Coefficients is Convergent

Let $\map S x$ be a trigonometric series: :$\map S x = \dfrac {a_0} 2 + \displaystyle \sum_{n \mathop = 1}^\infty \paren {a_n \cos n x + b_n \sin n x}$ Let the series: :$\displaystyle \sum_{n \mathop = 1}^\infty \paren {\size {a_n} + \size {b_n} }$ be convergent. Then $S$ is a convergent series.


Convergent Trigonometric Series is Periodic

Let $\map S x$ be a trigonometric series: :$\map S x = \dfrac {a_0} 2 + \displaystyle \sum_{n \mathop = 1}^\infty \paren {a_n \cos n x + b_n \sin n x}$ Let $S$ be convergent. Then $S$ is periodic: :$\forall r \in \Z: \map S {x + 2 r \pi} = \map S x$


Integral over 2 pi of Sine of m x by Sine of n x

Let $m, n \in \Z$ be integers. Let $\alpha \in \R$ be a real number. Then: :$\displaystyle \int_\alpha^{\alpha + 2 \pi} \sin m x \sin n x \rd x = \begin{cases} 0 & : m \ne n \\ \pi & : m = n \end{cases}$ That is: :$\displaystyle \int_\alpha^{\alpha + 2 \pi} \sin m x \sin n x \rd x = \pi \delta_{m n}$ where $\delta_{m n}$ is the Kronecker delta.


Integral over 2 pi of Cosine of m x by Cosine of n x

Let $m, n \in \Z$ be integers. Let $\alpha \in \R$ be a real number. Then: :$\ds \int_\alpha^{\alpha + 2 \pi} \cos m x \cos n x \rd x = \begin {cases} 0 & : m \ne n \\ \pi & : m = n \end {cases}$ That is: :$\ds \int_\alpha^{\alpha + 2 \pi} \cos m x \cos n x \rd x = \pi \delta_{m n}$ where $\delta_{m n}$ is the Kronecker delta.


Integral over 2 pi of Sine of m x by Cosine of n x

Let $m, n \in \Z$ be integers. Let $\alpha \in \R$ be a real number. Then: :$\displaystyle \int_\alpha^{\alpha + 2 \pi} \sin m x \cos n x \rd x = 0$


Integral over 2 pi of Sine of n x

Let $m \in \Z$ be an integer. Then: :$\displaystyle \int_\alpha^{\alpha + 2 \pi} \sin m x \rd x = 0$


Integral over 2 pi of Cosine of n x

Let $m \in \Z$ be an integer. Then: :$\displaystyle \int_\alpha^{\alpha + 2 \pi} \cos m x \rd x = \begin {cases} 0 & : m \ne 0 \\ 2 \pi & : m = 0 \end {cases}$


Coefficients of Sine Terms in Convergent Trigonometric Series

Let $\map S x$ be a trigonometric series which converges to $\map f x$ on the interval $\openint \alpha {\alpha + 2 \pi}$: :$\map f x = \dfrac {a_0} 2 + \displaystyle \sum_{m \mathop = 1}^\infty \left({a_m \cos m x + b_m \sin m x}\right)$ Then: :$\forall n \in \Z_{\ge 0}: b_n = \dfrac 1 \pi \displaystyle \int_\alpha^{\alpha + 2 \pi} \map f x \sin n x \rd x$


Sine of Angle plus Full Angle/Corollary

Let $n \in \Z$ be an integer. Then: :$\map \sin {x + 2 n \pi} = \sin x$


Cosine of Angle plus Full Angle/Corollary

Let $n \in \Z$ be an integer. Then: : $\cos \left({x + 2 n \pi}\right) = \cos x$


Fourier's Theorem

Let $\alpha \in \R$ be a real number. Let $\map f x$ be a real function which is defined and bounded on the interval $\openint \alpha {\alpha + 2 \pi}$. Let $f$ satisfy the Dirichlet conditions on $\openint \alpha {\alpha + 2 \pi}$: {{:Definition:Dirichlet Conditions}} Outside the interval $\openint \alpha {\alpha + 2 \pi}$, let $f$ be periodic and defined such that: :$\map f x = \map f {x + 2 \pi}$ Let $f$ be defined by the Fourier series: :$(1): \quad \displaystyle \frac {a_0} 2 + \sum_{n \mathop = 1}^\infty \paren {a_n \cos n x + b_n \sin n x}$ such that: :$\displaystyle a_n = \dfrac 1 \pi \int_\alpha^{\alpha + 2 \pi} \map f x \cos n x \rd x$ :$\displaystyle b_n = \dfrac 1 \pi \int_\alpha^{\alpha + 2 \pi} \map f x \sin n x \rd x$ Then for all $a \in \R$, $(1)$ converges to the sum: :$\displaystyle \frac 1 2 \paren {\lim_{x \mathop \to a^+} \map f x + \lim_{x \mathop \to a^-} \map f x}$ where the $\lim$ symbols denote the limit from the right and the limit from the left.


Definite Integral of Step Function

Let $\alpha, \beta \in \R$ be a real numbers such that $\alpha < \beta$. Let $f \left({x}\right)$ be a step function defined on the interval $\left[{\alpha \,.\,.\, \beta}\right]$: :$f \left({x}\right) = \lambda_1 \chi_{\mathbb I_1} + \lambda_2 \chi_{\mathbb I_2} + \cdots + \lambda_n \chi_{\mathbb I_n}$ where: :$\lambda_1, \lambda_2, \ldots, \lambda_n$ are real constants :$\mathbb I_1, \mathbb I_2, \ldots, \mathbb I_n$ are intervals, where these intervals partition $\left[{\alpha \,.\,.\, \beta}\right]$ :$\chi_{\mathbb I_1}, \chi_{\mathbb I_2}, \ldots, \chi_{\mathbb I_n}$ are characteristic functions of $\mathbb I_1, \mathbb I_2, \ldots, \mathbb I_n$. Then the definite integral of $f$ {{{WRT|Integration}} $x$ over $\left[{\alpha \,.\,.\, \beta}\right]$ is given by: :$\displaystyle \int_\alpha^\beta f \left({x}\right) \, \mathrm d x = \sum_{k \mathop = 1}^n \lambda_k \left({\beta_k - \alpha_k}\right)$ where $\alpha_k, \beta_k$ are the endpoints of $\mathbb I_k$ for $1 \le k \le n$.


Step Function satisfies Dirichlet Conditions

Let $\alpha, \beta \in \R$ be a real numbers such that $\alpha < \beta$. Let $\map f x$ be a step function defined on the interval $\openint \alpha \beta$. Then $f$ satisfies the Dirichlet conditions.


Parseval's Theorem/Formulation 1

Let $f$ be a real function which is square-integrable over the interval $\openint {-\pi} \pi$. Let $f$ be expressed by the Fourier series:  :$\map f x \sim \dfrac {a_0} 2 + \displaystyle \sum_{n \mathop = 1}^\infty \paren {a_n \cos n x + b_n \sin n x}$ Then: :$\displaystyle \frac 1 \pi \int_{-\pi}^\pi \size {\map {f^2} x} \rd x = \frac { {a_0}^2} 2 + \sum_{n \mathop = 1}^\infty \paren { {a_n}^2 + {b_n}^2}$