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Khachatour Koshtoyants (Armenian: Խաչատուր Սեդրակի Կոշտոյանց; Russian: Хачатур Седракович Коштоянц; September 26, 1900 – April 2, 1961) was a Soviet physiologist, Corresponding Member of the Academy of Sciences of the Soviet Union (since 1939), Member of the Armenian National Academy of Sciences (since 1943), Professor...	{ "page_id": 65277165, "title": "Khachatour Koshtoyants" }
The cyclol hypothesis is the now discredited first structural model of a folded, globular protein, formulated in the 1930s. It was based on the cyclol reaction of peptide bonds proposed by physicist Charles Frank in 1936, in which two peptide groups are chemically crosslinked. These crosslinks are covalent analogs of t...	{ "page_id": 6884590, "title": "Cyclol" }
subunits, now known as a change in quaternary structure. The chemical structure of proteins was still under debate at that time. The most accepted (and ultimately correct) hypothesis was that proteins are linear polypeptides, i.e., unbranched polymers of amino acids linked by peptide bonds. However, a typical protein i...	{ "page_id": 6884590, "title": "Cyclol" }
not exclude the possibility that denaturation corresponded to a chemical change in the protein structure, a hypothesis that was considered a (distant) possibility until the 1950s. X-ray crystallography had just begun as a discipline in 1911, and had advanced relatively rapidly from simple salt crystals to crystals of c...	{ "page_id": 6884590, "title": "Cyclol" }
in understanding protein structure. == Basic theory == Wrinch developed this suggestion into a full-fledged model of protein structure. The basic cyclol model was laid out in her first paper (1936). She noted the possibility that polypeptides might cyclize to form closed rings (true) and that these rings might form int...	{ "page_id": 6884590, "title": "Cyclol" }
was untenable as a model for globular proteins. == Stabilizing energies == In two tandem Letters to the Editor (1936), Wrinch and Frank addressed the question of whether the cyclol form of the peptide group was indeed more stable than the amide form. A relatively simple calculation showed that the cyclol form is signif...	{ "page_id": 6884590, "title": "Cyclol" }
the amino-acid side chains. Wrinch proposed that steric complementarity was one of chief factors in determining whether a small molecule would bind to a protein. Wrinch speculated that proteins are responsible for the synthesis of all biological molecules. Noting that cells digest their proteins only under extreme star...	{ "page_id": 6884590, "title": "Cyclol" }
angle δ = arccos(-1/3) ≈ 109.47°. A large variety of closed polyhedra meeting this criterion can be constructed, of which the simplest are the truncated tetrahedron, the truncated octahedron, and the octahedron, which are Platonic solids or semiregular polyhedra. Considering the first series of "closed cyclols" (those ...	{ "page_id": 6884590, "title": "Cyclol" }
their own synthesis, analogous to the Watson-Francis Crick concept of DNA templating its own replication. Given that many biological molecules such as sugars and sterols have a hexagonal structure, it was plausible to assume that their synthesizing proteins likewise had a hexagonal structure. Wrinch summarized her mode...	{ "page_id": 6884590, "title": "Cyclol" }
5). Haurowitz showed chemically that the outside of proteins could not have a large number of hydroxyl groups, a key prediction of the cyclol model, whereas Meyer and Hohenemser showed that cyclol condensations of amino acids did not exist even in minute quantities as a transition state. More general chemical arguments...	{ "page_id": 6884590, "title": "Cyclol" }
the cyclol model on that basis alone. In reply to the chemical criticisms, Wrinch suggested that the model compounds and simple bimolecular reactions studied need not pertain to the cyclol model, and that steric hindrance may have prevented the surface hydroxyl groups from reacting. On the residue-number criticism, Wri...	{ "page_id": 6884590, "title": "Cyclol" }
called an azacyclol. By analogy, an oxacyclol is formed when an OH hydroxyl group is added to a peptidyl carbonyl group. Likewise, a thiacyclol is formed by adding an SH thiol moiety to a peptidyl carbonyl group. The oxacyclol alkaloid ergotamine from the fungus Claviceps purpurea was the first identified cyclol. The c...	{ "page_id": 6884590, "title": "Cyclol" }
Project SHAD, an acronym for Shipboard Hazard and Defense, was part of a larger effort called Project 112, which was conducted during the 1960s. Project SHAD encompassed tests designed to identify U.S. warships' vulnerabilities to attacks with chemical agents or biological warfare agents and to develop procedures to re...	{ "page_id": 1445107, "title": "Project SHAD" }
actually completed. == Declassification == Public Law 107–314 required the identification and release of not only Project 112 information to the United States Veterans Administration, but also that of any other projects or tests where a veteran might have been exposed to a chemical or biological warfare agent, and dire...	{ "page_id": 1445107, "title": "Project SHAD" }
he believes the Pentagon used him and other service members to test weapons, and that those tests included agents, vaccines, and decontamination products which have led to serious medical problems, including cancer. Secrecy agreements can now be ignored by veterans in order to pursue healthcare concerns within the Depa...	{ "page_id": 1445107, "title": "Project SHAD" }
USNS Silas Bent (T-AGS-26) USS Tioga County (LST-1158) USS Tiru (SS-416) USS Wexford County (LST-1168) US Navy Lighter Barge (YFN-811) US Army Large Tug LT-2080 US Army Large Tug LT-2081 US Army Large Tug LT-2085 US Army Large Tug LT-2086 US Army Large Tug LT-2087 US Army Large Tug LT-2088 === Air units === Meteorologi...	{ "page_id": 1445107, "title": "Project SHAD" }
SHAD -- AUTUMN GOLD, May 1964 PROJECT SHAD -- BIG JACK PROJECT SHAD -- NIGHT TRAIN, December 1964	{ "page_id": 1445107, "title": "Project SHAD" }
The copula linguae or copula, is a swelling that forms from the second pharyngeal arch, late in the fourth week of embryogenesis. During the fifth and sixth weeks the copula becomes overgrown and covered by the hypopharyngeal eminence which forms mostly from the third pharyngeal arch and in part from the fourth pharyng...	{ "page_id": 7998715, "title": "Copula linguae" }
Fungal DNA barcoding is the process of identifying species of the biological kingdom Fungi through the amplification and sequencing of specific DNA sequences and their comparison with sequences deposited in a DNA barcode database such as the ISHAM reference database, or the Barcode of Life Data System (BOLD). In this a...	{ "page_id": 62065916, "title": "Fungal DNA barcoding" }
records at a high taxonomic level of identification, queries – even when they might have a close or exact match in the reference database – will not be informative if the closest match is only identified to phylum or class level. Another crucial prerequisite for DNA barcoding is the ability to unambiguously trace the p...	{ "page_id": 62065916, "title": "Fungal DNA barcoding" }
conserved DNA sequence, as they code for structural parts of the ribosome, which is a key component in intracellular protein synthesis. Due to several advantages of ITS (see below) and a comprehensive amount of sequence data accumulated in the 1990s and early 2000s, Begerow et al. (2010) and Schoch et al. (2012) propos...	{ "page_id": 62065916, "title": "Fungal DNA barcoding" }
rate varies greatly among fungal groups, from 65% in non-Dikarya (including the now paraphyletic Mucoromycotina, the Chytridiomycota and the Blastocladiomycota) to 100% in Saccharomycotina and Basidiomycota (with the exception of very low success in Pucciniomycotina). Furthermore, the choice of primers for ITS amplific...	{ "page_id": 62065916, "title": "Fungal DNA barcoding" }
Rhizoctonia solani (n=608), or even 24.75% in Pisolithus tinctorius (n=113). In cases of high intraspecific ITS variability, the application of a threshold of 3% sequence variability – a canonical upper value for intraspecific variation – will therefore lead to a higher estimate of operational taxonomic units (OTUs), i...	{ "page_id": 62065916, "title": "Fungal DNA barcoding" }
similar to the situation in plants, where the plastidial genes rbcL, matK and trnH-psbA, as well as the nuclear ITS are often used in combination for DNA barcoding. === Translational elongation factor 1α (TEF1α) – the secondary fungal barcode === The translational elongation factor 1α is part of the eucaryotic elongati...	{ "page_id": 62065916, "title": "Fungal DNA barcoding" }
with the sequence 5'-ACHGTRCCRATACCACCRATCTT-3'. In addition, a number of new primers was developed, with the primer pair in bold resulting in a high average amplification success of 88%: Primers used for the investigation of Rhizophydiales and especially Batrachochytrium dendrobatidis, a pathogen of amphibia, are the ...	{ "page_id": 62065916, "title": "Fungal DNA barcoding" }
Primers ==== For basidiomycetous yeasts, the forward primer F63 with the sequence 5'-GCATATCAATAAGCGGAGGAAAAG-3', and the reverse primer LR3 with the sequence 5'-GGTCCGTGTTTCAAGACGG-3' have been successfully used for PCR amplification of the D1/D23 domain. The D1/D2 domain of ascomycetous yeasts like Candida can be amp...	{ "page_id": 62065916, "title": "Fungal DNA barcoding" }
been successfully used for the differentiation of strains of Xanthophyllomyces dendrorhous as well as for species distinction in the psychrophilic genus Mrakia (Cystofilobasidiales). Due to these results, IGS has been recommended as a genetic marker for additional differentiation (along with D1/D2 and ITS) of closely r...	{ "page_id": 62065916, "title": "Fungal DNA barcoding" }
DNA barcode candidate by Lewis et al. (2011) based on proteome data, with the developed universal primer pair being subsequently tested on actual samples by Stielow et al. (2015). The forward primer TOP1_501-F with the sequence 5'-TGTAAAACGACGGCCAGT-ACGAT-ACTGCCAAGGTTTTCCGTACHTACAACGC-3' (where the first section marks ...	{ "page_id": 62065916, "title": "Fungal DNA barcoding" }
the two databases (GenBank and UNITE) used for ITS sequence comparison gave different identification results in some of the samples. Correct labelling of mushrooms intended for consumption was also investigated by Raja et al. (2016), who used the ITS region for DNA barcoding from dried mushrooms, mycelium powders, and ...	{ "page_id": 62065916, "title": "Fungal DNA barcoding" }
a wide range of localised to invasive diseases: ITS could not distinguish between S. apiospermum and S. boydii, whereas with TEF1α all investigated species of this genus could be accurately identified. This study therefore underlines the usefulness of applying more than one DNA barcoding marker for fungal species ident...	{ "page_id": 62065916, "title": "Fungal DNA barcoding" }
Charmed baryons are a category of composite particles comprising all baryons made of at least one charm quark. Since their first observation in the 1970s, a large number of distinct charmed baryon states have been identified. Observed charmed baryons have masses ranging between 2300 and 2700 MeV/c2. In 2002, the SELEX ...	{ "page_id": 3214588, "title": "Charmed baryon" }
indicated by subscripts. For example, a Ξ+cb is made of one bottom, one charmed quark, and it can be deduced from the charge of the charm (+⁠2/3⁠e) and bottom quark (−⁠1/3⁠e) that the other quark must be an up quark (+⁠2/3⁠e). Sometimes asterisks or primes are used to indicate a resonance. == Properties == The importan...	{ "page_id": 3214588, "title": "Charmed baryon" }
in addition to a smooth "phase space" background. The width of the peak will be governed by the resolution of the detector, provided that the charmed baryon is reasonably stable (such as the Λ+c which has a lifetime of around (2±10)×10−13 s). Other, higher states of charmed baryons, which decay by the strong interactio...	{ "page_id": 3214588, "title": "Charmed baryon" }
weak interaction, taking into account the large available phase space. The lifetime measurement has contributions from a number of experiments, notably FOCUS, SELEX and CLEO. === Decays === The Λ+c decays into a multitude of different final states, according to the rules of weak decays. The decay into a proton, kaon an...	{ "page_id": 3214588, "title": "Charmed baryon" }
The situation is directly analogous to the strange baryon nomenclature. The ground state (that is, with no orbital angular momentum) baryons can also be pictured thus. Each quark is a spin 1/2 particle. The spins can be pointed up, or down. In Λ+c ground state, the two light quarks point up-down to give a zero spin diq...	{ "page_id": 3214588, "title": "Charmed baryon" }
charged state from the Columbia–Brookhaven collaboration. In 1987–89, a series of experiments (E-400 at Fermilab, ARGUS and CLEO) with much larger statistics, found clear evidence for both the doubly charged and neutral states (though the E-400 neutral state turned out to be a false signal). It became clear that the ma...	{ "page_id": 3214588, "title": "Charmed baryon" }
working at CERN. They found a significant peak in the decay mode ΛK−π+π+ at a mass of 2460±25 MeV/c2. The present value for the mass is taken from an average of 6 experiments, and is 2467.9±0.4 MeV/c2. The Ξ0c was discovered in 1989 by the CLEO, who measured a peak in the decay mode of Ξ−π+ with a mass of 2471±5 MeV/c2...	{ "page_id": 3214588, "title": "Charmed baryon" }
necessary studies to be able to quote a mass with an uncertainty. Therefore, though there is no doubt that the particle has been discovered, there is no definitive measurement of its mass. == References ==	{ "page_id": 3214588, "title": "Charmed baryon" }
Lilabati Bhattacharjee (née Ray) was a mineralogist, crystallographer and a physicist. She studied with the scientist Satyendra Nath Bose, and completed her MSc in physics from the University College of Science and Technology (commonly known as Rajabazar Science College), University of Calcutta in 1951. Mrs Bhattacharj...	{ "page_id": 53349635, "title": "Lilabati Bhattacharjee" }
A fruit press is a device used to separate fruit solids—stems, skins, seeds, pulp, leaves, and detritus—from fruit juice. == History == In the United States, Madeline Turner invented the Turner's Fruit-Press, in 1916. == Cider press == A cider press is used to crush apples or pears. In North America, the unfiltered jui...	{ "page_id": 68869, "title": "Fruit press" }
passes through them; the axle is joined to the vertical axis; its other end juts out from the trough; a horse → is harnessed to it; the ← horse → pulls the axle by walking round the trough, which also moves the pressing stones in the trough where the apples are pounded. When they are judged to be sufficiently crushed, ...	{ "page_id": 68869, "title": "Fruit press" }
press grapes during wine making. == Oil press == An oil press is a device used to extract oil from plants and fruits. == DIY fruit press == Given the simplicity of the design, and high usability, some people (e.g. those owning an orchard) have started building their own do-it-yourself (DIY) fruit press and have uploade...	{ "page_id": 68869, "title": "Fruit press" }
BioSentinel is a lowcost CubeSat spacecraft on a astrobiology mission that will use budding yeast to detect, measure, and compare the impact of deep space radiation on DNA repair over long time beyond low Earth orbit. Selected in 2013 for a 2022 launch, the spacecraft will operate in the deep space radiation environmen...	{ "page_id": 46796038, "title": "BioSentinel" }
deleterious lesions generated by ionizing radiation. Budding yeast was selected not only because of its flight heritage, but also because of its similarities with human cells, especially its DSB repair mechanisms. The biosensor consists of specifically engineered yeast strains and growth medium containing a metabolic i...	{ "page_id": 46796038, "title": "BioSentinel" }
vehicle from which it is deployed to a lunar flyby trajectory and into an Earth-trailing heliocentric orbit. Of the total 6 Units volume, 4 Units will hold the science payload, including a radiation dosimeter and a dedicated 3-color spectrometer for each well; 0.5U will house the ADCS (Attitude Determination and Contro...	{ "page_id": 46796038, "title": "BioSentinel" }
Reason LLC, Tampa, Florida The 3 CubeSat missions removed from Artemis 1 Lunar Flashlight will map exposed water ice on the Moon Cislunar Explorers, Cornell University, Ithaca, New York Earth Escape Explorer (CU-E3), University of Colorado Boulder Astrobiology missions Bion BIOPAN Biosatellite program List of microorga...	{ "page_id": 46796038, "title": "BioSentinel" }
In theoretical physics, the Pasterski–Strominger–Zhiboedov (PSZ) triangle or infrared triangle is a series of relationships between three groups of concepts involving the theory of relativity, quantum field theory and quantum gravity. The triangle highlights connections already known or demonstrated by its authors, Sab...	{ "page_id": 75500805, "title": "Pasterski–Strominger–Zhiboedov triangle" }
effects; vacuum transitions tie together asymptotic symmetries and memory effects; Ward's identities tie together soft theorems and asymptotic symmetries. So, for example: the soft graviton theorem (a.1) is related to the supertranslations (b.1) by a Ward's identity; the supertranslations (b.1) correspond to different ...	{ "page_id": 75500805, "title": "Pasterski–Strominger–Zhiboedov triangle" }
Foundations of the Czech chemical nomenclature (Czech: české chemické názvosloví) and terminology were laid during the 1820s and 1830s. These early naming conventions fit the Czech language and, being mostly the work of a single person, Jan Svatopluk Presl, provided a consistent way to name chemical compounds. Over tim...	{ "page_id": 14617864, "title": "Czech chemical nomenclature" }
they represented. Originally there were five suffixes: -ný, -natý, -itý, -ový, and -elý. These were later expanded to eight by Vojtěch Šafařík: -ný, -natý, -itý, -ičitý, -ičný and -ečný, -ový, -istý, and -ičelý, representing oxidation numbers from 1 to 8. For example, železnatý corresponds to 'ferrous' and železitý to ...	{ "page_id": 14617864, "title": "Czech chemical nomenclature" }
The Czechoslovak Academy of Sciences, founded in 1953, took over responsibility for maintenance of the nomenclature and proper implementation of the IUPAC recommendations. Since the Velvet Revolution (1989) this activity has slowed down considerably. == Oxidation state suffixes == == Notes == == External links == Websi...	{ "page_id": 14617864, "title": "Czech chemical nomenclature" }
Arcanadea (アルカナディア, Arukanadia) is a Japanese series of heavily customizable plastic model kit girls created and produced by Kotobukiya, released since December 2021, with necömi as the character designer. An anime television series adaptation has been announced. == Characters == Lumitea (ルミティア, Rumitia) Voiced by: Kae...	{ "page_id": 78646537, "title": "Arcanadea" }
Congelation (from Latin: congelātiō, lit. 'freezing, congealing') was a term used in medieval and early modern alchemy for the process known today as crystallization. In the Secreta alchymiae ('The Secret of Alchemy') attributed to Khalid ibn Yazid (c. 668–704 or 709), it is one of "the four principal operations", alon...	{ "page_id": 2428176, "title": "Congelation" }
Foxfire, also called fairy fire and chimpanzee fire, is the bioluminescence created by some species of fungi present in decaying wood. The bluish-green glow is attributed to a luciferase, an oxidative enzyme, which emits light as it reacts with a luciferin. The phenomenon has been known since ancient times, with its so...	{ "page_id": 1248534, "title": "Foxfire" }
meaning "false", rather than from the name of the animal. The association of foxes with such lights is widespread, however, and occurs also in Japanese folklore in the form of kitsune-bi (狐火 lit. 'Fox's fire/flame'). == See also == Aurora Borealis, called "revontulet" (literally "foxfires") in the Finnish language List...	{ "page_id": 1248534, "title": "Foxfire" }
Giomar Helena Borrero-Pérez is a Colombian marine biologist. In 2012 she became the sixth Colombian scientist to be awarded a L'Oréal-UNESCO For Women in Science Award. Her work considers the conservation of sea cucumbers. == Early life and education == Borrero was born in Mitú. She attended the National Indigenous Nor...	{ "page_id": 62393626, "title": "Giomar Helena Borrero-Pérez" }
Filters in topology, a subfield of mathematics, can be used to study topological spaces and define all basic topological notions such as convergence, continuity, compactness, and more. Filters, which are special families of subsets of some given set, also provide a common framework for defining various types of limits ...	{ "page_id": 47516955, "title": "Filters in topology" }
{S}}\geq {\mathcal {B}},} which denotes B ≤ S {\displaystyle {\mathcal {B}}\leq {\mathcal {S}}} and is expressed by saying that S {\displaystyle {\mathcal {S}}} is subordinate to B , {\displaystyle {\mathcal {B}},} also establishes a relationship in which S {\displaystyle {\mathcal {S}}} is to B {\displaystyle {\mathca...	{ "page_id": 47516955, "title": "Filters in topology" }
subordinate filters and subnets because as detailed below, there exist subordinate filters whose filter/subordinate-filter relationship cannot be described in terms of the corresponding net/subnet relationship; this issue can however be resolved by using a less commonly encountered definition of "subnet", which is that...	{ "page_id": 47516955, "title": "Filters in topology" }
x} (and of anything else that it contains); Does not contain the empty set: ∅ ∉ B {\displaystyle \varnothing \not \in {\mathcal {B}}} – just as no neighborhood of x {\displaystyle x} is empty; Closed under finite intersections: If B , C ∈ B then B ∩ C ∈ B {\displaystyle B,C\in {\mathcal {B}}{\text{ then }}B\cap C\in {\...	{ "page_id": 47516955, "title": "Filters in topology" }
and filters, which never fail to characterize topological properties. Nets directly generalize the notion of a sequence since nets are, by definition, maps I → X {\displaystyle I\to X} from an arbitrary directed set ( I , ≤ ) {\displaystyle (I,\leq )} into the space X . {\displaystyle X.} A sequence is just a net whose...	{ "page_id": 47516955, "title": "Filters in topology" }
5 , … } x ≥ 3 = { x 3 , x 4 , x 5 , x 6 , … } ⋮ x ≥ n = { x n , x n + 1 , x n + 2 , x n + 3 , … } ⋮ {\displaystyle {\begin{alignedat}{8}x_{\geq 1}=\;&\{&&x_{1},&&x_{2},&&x_{3},&&x_{4},&&\ldots &&\,\}\\[0.3ex]x_{\geq 2}=\;&\{&&x_{2},&&x_{3},&&x_{4},&&x_{5},&&\ldots &&\,\}\\[0.3ex]x_{\geq 3}=\;&\{&&x_{3},&&x_{4},&&x_{5},...	{ "page_id": 47516955, "title": "Filters in topology" }
but it does "generate" a filter via taking its upward closure (which consists of all supersets of all tails). The same is true of other important families of sets such as any neighborhood basis at a given point, which in general is also not a filter but does generate a filter via its upward closure (in particular, it g...	{ "page_id": 47516955, "title": "Filters in topology" }
defined in terms of composition of functions rather than sets and it is more readily applied to functions like nets than to sets like filters (a prominent example of an inverse limit is the Cartesian product). Filters may be awkward to use in certain situations, such as when switching between a filter on a space X {\di...	{ "page_id": 47516955, "title": "Filters in topology" }
applied to the sets that constitute a filter; an example of such a result might be one of continuity's characterizations in terms of preimages of open/closed sets or in terms of the interior/closure operators. Special types of filters called ultrafilters have many useful properties that can significantly help in provin...	{ "page_id": 47516955, "title": "Filters in topology" }
of ℘ ( X ) . {\displaystyle \wp (X).} Families of sets will be denoted by upper case calligraphy letters such as B {\displaystyle {\mathcal {B}}} , C {\displaystyle {\mathcal {C}}} , and F {\displaystyle {\mathcal {F}}} . Whenever these assumptions are needed, then it should be assumed that X {\displaystyle X} is non-e...	{ "page_id": 47516955, "title": "Filters in topology" }
⊆ ℘ ( X ) {\displaystyle {\mathcal {B}}\subseteq \wp (X)} is and similarly the downward closure of B {\displaystyle {\mathcal {B}}} is B ↓ := { S ⊆ B : B ∈ B } = ⋃ B ∈ B ℘ ( B ) . {\displaystyle {\mathcal {B}}^{\downarrow }:=\{S\subseteq B~:~B\in {\mathcal {B}}\,\}={\textstyle \bigcup \limits _{B\in {\mathcal {B}}}}\wp...	{ "page_id": 47516955, "title": "Filters in topology" }
{\displaystyle i<j} is defined to mean that i ≤ j {\displaystyle i\leq j} holds but it is not true that j ≤ i {\displaystyle j\leq i} (if ≤ {\displaystyle \,\leq \,} is antisymmetric then this is equivalent to i ≤ j and i ≠ j {\displaystyle i\leq j{\text{ and }}i\neq j} ). A net in X {\displaystyle X} is a map from a n...	{ "page_id": 47516955, "title": "Filters in topology" }
\left\{x_{\geq i}~:~i\in I\right\}} and it is for this reason that in general, when dealing with the prefilter of tails of a net, the strict inequality < {\displaystyle \,<\,} may not be used interchangeably with the inequality ≤ . {\displaystyle \,\leq .} === Filters and prefilters === The following is a list of prope...	{ "page_id": 47516955, "title": "Filters in topology" }
℘ ( X ) {\displaystyle \wp (X)} is also called the degenerate filter on X {\displaystyle X} (despite not actually being a filter). It is the only dual ideal on X {\displaystyle X} that is not a filter on X . {\displaystyle X.} If ( X , τ ) {\displaystyle (X,\tau )} is a topological space and x ∈ X , {\displaystyle x\in...	{ "page_id": 47516955, "title": "Filters in topology" }
= ( x i ) i = 1 ∞ . {\displaystyle x_{\bullet }=\left(x_{i}\right)_{i=1}^{\infty }.} B {\displaystyle {\mathcal {B}}} is an elementary filter or a sequential filter on X {\displaystyle X} if B {\displaystyle {\mathcal {B}}} is a filter on X {\displaystyle X} generated by some elementary prefilter. The filter of tails g...	{ "page_id": 47516955, "title": "Filters in topology" }
family F ⊆ Filters ⁡ ( X ) {\displaystyle \mathbb {F} \subseteq \operatorname {Filters} (X)} is itself a filter on X {\displaystyle X} called the infimum or greatest lower bound of F in Filters ⁡ ( X ) , {\displaystyle \mathbb {F} {\text{ in }}\operatorname {Filters} (X),} which is why it may be denoted by ⋀ F ∈ F F . ...	{ "page_id": 47516955, "title": "Filters in topology" }
{\mathcal {S}}\leq {\mathcal {B}}{\text{ and }}{\mathcal {S}}\leq {\mathcal {F}}} then necessarily S ≤ B ( ∪ ) F . {\displaystyle {\mathcal {S}}\leq {\mathcal {B}}\,(\cup )\,{\mathcal {F}}.} More generally, if B and F {\displaystyle {\mathcal {B}}{\text{ and }}{\mathcal {F}}} are non−empty families and if S := { S ⊆ ℘ ...	{ "page_id": 47516955, "title": "Filters in topology" }
:= { F 1 ∩ ⋯ ∩ F n : n ∈ N and every F i belongs to some F ∈ F } {\displaystyle \pi \left(\cup \mathbb {F} \right):=\left\{F_{1}\cap \cdots \cap F_{n}~:~n\in \mathbb {N} {\text{ and every }}F_{i}{\text{ belongs to some }}{\mathcal {F}}\in \mathbb {F} \right\}} is the π-system generated by ∪ F . {\displaystyle \cup \mat...	{ "page_id": 47516955, "title": "Filters in topology" }
exists then necessarily ⋁ F ∈ F F = π ( ∪ F ) ↑ X {\displaystyle {\textstyle \bigvee \limits _{{\mathcal {F}}\in \mathbb {F} }}{\mathcal {F}}=\pi \left(\cup \mathbb {F} \right)^{\uparrow X}} (as defined above) and ⋁ F ∈ F F {\displaystyle {\textstyle \bigvee \limits _{{\mathcal {F}}\in \mathbb {F} }}{\mathcal {F}}} wil...	{ "page_id": 47516955, "title": "Filters in topology" }
B ( ∩ ) F {\displaystyle {\mathcal {B}}\,(\cap )\,{\mathcal {F}}} is a prefilter (resp. a filter) if and only if it is non-degenerate (or said differently, if and only if B and F {\displaystyle {\mathcal {B}}{\text{ and }}{\mathcal {F}}} mesh), in which case it is one of the coarsest prefilters (resp. the coarsest filt...	{ "page_id": 47516955, "title": "Filters in topology" }
B ↑ X = { S ⊆ X : p ∈ S } = { { p } ∪ T : T ⊆ { 1 , 2 , 3 } } . {\displaystyle {\mathcal {B}}^{\uparrow X}=\{S\subseteq X:p\in S\}=\{\{p\}\cup T~:~T\subseteq \{1,2,3\}\}.} All three of B , {\displaystyle {\mathcal {B}},} the π-system B {\displaystyle {\mathcal {B}}} generates, and B ↑ X {\displaystyle {\mathcal {B}}^{\...	{ "page_id": 47516955, "title": "Filters in topology" }
prefilter that is finer than B . {\displaystyle {\mathcal {B}}.} If X = R n {\displaystyle X=\mathbb {R} ^{n}} (with 1 ≤ n ∈ N {\displaystyle 1\leq n\in \mathbb {N} } ) then the set B LebFinite {\displaystyle {\mathcal {B}}_{\operatorname {LebFinite} }} of all B ∈ B {\displaystyle B\in {\mathcal {B}}} such that B {\dis...	{ "page_id": 47516955, "title": "Filters in topology" }
for compact Hausdorff spaces and the Alexander subbase theorem) and in functional analysis (such as the Hahn–Banach theorem) can be proven using only the ultrafilter lemma; the full strength of the axiom of choice might not be needed. ==== Kernels ==== The kernel is useful in classifying properties of prefilters and ot...	{ "page_id": 47516955, "title": "Filters in topology" }
{B}}.} Family of examples: For any non-empty C ⊆ R , {\displaystyle C\subseteq \mathbb {R} ,} the family B C = { R ∖ ( r + C ) : r ∈ R } {\displaystyle {\mathcal {B}}_{C}=\{\mathbb {R} \setminus (r+C)~:~r\in \mathbb {R} \}} is free but it is a filter subbase if and only if no finite union of the form ( r 1 + C ) ∪ ⋯ ∪ ...	{ "page_id": 47516955, "title": "Filters in topology" }
in addition X {\displaystyle X} is finite, then there are no ultrafilters on X {\displaystyle X} other than these. The next theorem shows that every ultrafilter falls into one of two categories: either it is free or else it is a principal filter generated by a single point. === Finer/coarser, subordination, and meshing...	{ "page_id": 47516955, "title": "Filters in topology" }
B on S , {\displaystyle {\mathcal {B}}{\text{ on }}S,} which is the family B \| S = { B ∩ S : B ∈ B } , {\displaystyle {\mathcal {B}}{\big \vert }_{S}=\{B\cap S~:~B\in {\mathcal {B}}\},} does not contain the empty set, where the trace is also called the restriction of B to S . {\displaystyle {\mathcal {B}}{\text{ to }}S...	{ "page_id": 47516955, "title": "Filters in topology" }
x i n + 1 , … } , {\displaystyle x_{i_{\geq n}}=\left\{x_{i_{n}},x_{i_{n+1}},\ldots \right\},} it is sufficient to have i ≤ i n . {\displaystyle i\leq i_{n}.} Since i 1 < i 2 < ⋯ {\displaystyle i_{1}<i_{2}<\cdots } are strictly increasing integers, there exists n ∈ N {\displaystyle n\in \mathbb {N} } such that i n ≥ i ...	{ "page_id": 47516955, "title": "Filters in topology" }
family that is coarser than a filter subbase must itself be a filter subbase. Every filter subbase is coarser than both the π-system that it generates and the filter that it generates. If C and F {\displaystyle {\mathcal {C}}{\text{ and }}{\mathcal {F}}} are families such that C ≤ F , {\displaystyle {\mathcal {C}}\leq ...	{ "page_id": 47516955, "title": "Filters in topology" }
Two upward closed (in X {\displaystyle X} ) subsets of ℘ ( X ) {\displaystyle \wp (X)} are equivalent if and only if they are equal. If B ⊆ ℘ ( X ) {\displaystyle {\mathcal {B}}\subseteq \wp (X)} then necessarily ∅ ≤ B ≤ ℘ ( X ) {\displaystyle \varnothing \leq {\mathcal {B}}\leq \wp (X)} and B {\displaystyle {\mathcal ...	{ "page_id": 47516955, "title": "Filters in topology" }
is the trivial filter. In general, this conclusion of equality does not extend to non−trivial filters (one exception is when both families are filters). Meshes with F {\displaystyle {\mathcal {F}}} Is finer than F {\displaystyle {\mathcal {F}}} Is coarser than F {\displaystyle {\mathcal {F}}} Is equivalent to F {\displ...	{ "page_id": 47516955, "title": "Filters in topology" }
the filter that it generates. This is why prefilters can, by and large, be used interchangeably with the filters that they generate while filter subbases cannot. == Set theoretic properties and constructions relevant to topology == === Trace and meshing === If B {\displaystyle {\mathcal {B}}} is a prefilter (resp. filt...	{ "page_id": 47516955, "title": "Filters in topology" }
{\mathcal {B}}.} Then B and S {\displaystyle {\mathcal {B}}{\text{ and }}S} mesh and B ∪ { S } {\displaystyle {\mathcal {B}}\cup \{S\}} generates a filter on X {\displaystyle X} that is strictly finer than B . {\displaystyle {\mathcal {B}}.} When prefilters mesh Given non-empty families B and C , {\displaystyle {\mathc...	{ "page_id": 47516955, "title": "Filters in topology" }
characterization of "mesh" entirely in terms of subordination (that is, ≤ {\displaystyle \,\leq \,} ): Two prefilters (resp. filter subbases) B and C {\displaystyle {\mathcal {B}}{\text{ and }}{\mathcal {C}}} mesh if and only if there exists a prefilter (resp. filter subbase) F {\displaystyle {\mathcal {F}}} such that ...	{ "page_id": 47516955, "title": "Filters in topology" }
( g ( B ) ) . {\displaystyle g({\mathcal {B}}){\text{ and }}g^{-1}(g({\mathcal {B}})).} The image under a map f : X → Y {\displaystyle f:X\to Y} of an ultra set B ⊆ ℘ ( X ) {\displaystyle {\mathcal {B}}\subseteq \wp (X)} is again ultra and if B {\displaystyle {\mathcal {B}}} is an ultra prefilter then so is f ( B ) . {...	{ "page_id": 47516955, "title": "Filters in topology" }
Y ) . {\displaystyle {\mathcal {B}}\subseteq \wp (Y).} Under the assumption that f : X → Y {\displaystyle f:X\to Y} is surjective: f − 1 ( B ) {\displaystyle f^{-1}({\mathcal {B}})} is a prefilter (resp. filter subbase, π-system, closed under finite unions, proper) if and only if this is true of B . {\displaystyle {\ma...	{ "page_id": 47516955, "title": "Filters in topology" }
X ) {\displaystyle f:X\to f(X)} can be used in place of f : X → Y . {\displaystyle f:X\to Y.} For example: f − 1 ( B ) {\displaystyle f^{-1}({\mathcal {B}})} is a prefilter (resp. filter subbase, π-system, proper) if and only if this is true of B \| f ( X ) . {\displaystyle {\mathcal {B}}{\big \vert }_{f(X)}.} In this w...	{ "page_id": 47516955, "title": "Filters in topology" }
to In − 1 ⁡ ( B ) . {\displaystyle \operatorname {In} ^{-1}({\mathcal {B}}).} This observation allows the results in this subsection to be applied to investigating the trace on a set. ==== Subordination is preserved by images and preimages ==== The relation ≤ {\displaystyle \,\leq \,} is preserved under both images and...	{ "page_id": 47516955, "title": "Filters in topology" }
Products of prefilters === Suppose X ∙ = ( X i ) i ∈ I {\displaystyle X_{\bullet }=\left(X_{i}\right)_{i\in I}} is a family of one or more non-empty sets, whose product will be denoted by ∏ X ∙ := ∏ i ∈ I X i , {\displaystyle {\textstyle \prod _{}}X_{\bullet }:={\textstyle \prod \limits _{i\in I}}X_{i},} and for every ...	{ "page_id": 47516955, "title": "Filters in topology" }
a filter subbase then the family ⋃ i ∈ I Pr X i − 1 ( B i ) {\displaystyle {\textstyle \bigcup \limits _{i\in I}}\Pr {}_{X_{i}}^{-1}\left({\mathcal {B}}_{i}\right)} is a filter subbase for the filter on ∏ X ∙ {\displaystyle {\textstyle \prod }X_{\bullet }} generated by B ∙ . {\displaystyle {\mathcal {B}}_{\bullet }.} I...	{ "page_id": 47516955, "title": "Filters in topology" }
filter on the codomain, although it will be a prefilter. The situation is the same with preimages under non-injective maps (even if the map is surjective). If S ⊆ X {\displaystyle S\subseteq X} is a proper subset then any filter on S {\displaystyle S} will not be a filter on X , {\displaystyle X,} although it will be a...	{ "page_id": 47516955, "title": "Filters in topology" }
subset of F i . {\displaystyle F_{i}.} The same is not true going "upward", for if F 0 = X ∈ F {\displaystyle F_{0}=X\in {\mathcal {F}}} then there is no set in F {\displaystyle {\mathcal {F}}} that contains X {\displaystyle X} as a proper subset. Thus when it comes to limiting behavior (which is a topic central to the...	{ "page_id": 47516955, "title": "Filters in topology" }
to a point or subset x {\displaystyle x} if and only if it is finer than the neighborhood filter at x . {\displaystyle x.} A family B {\displaystyle {\mathcal {B}}} converging to a point x {\displaystyle x} may be indicated by writing B → x or lim B → x in X {\displaystyle {\mathcal {B}}\to x{\text{ or }}\lim {\mathcal...	{ "page_id": 47516955, "title": "Filters in topology" }
S τ ( s ) {\displaystyle \tau (S)={\textstyle \bigcap \limits _{s\in S}}\tau (s)} when S ≠ ∅ {\displaystyle S\neq \varnothing } ). Examples If X := R n {\displaystyle X:=\mathbb {R} ^{n}} is Euclidean space and ‖ x ‖ {\displaystyle \\|x\\|} denotes the Euclidean norm (which is the distance from the origin, defined as usu...	{ "page_id": 47516955, "title": "Filters in topology" }
{\displaystyle r} ranging over 1 , 1 / 2 , 1 / 3 , 1 / 4 , … {\displaystyle 1,\,1/2,\,1/3,\,1/4,\ldots } (or over any other positive decreasing sequence) instead of over all positive reals. Drawing or imagining any one of these sequences of sets when X = R 2 {\displaystyle X=\mathbb {R} ^{2}} has dimension n = 2 {\disp...	{ "page_id": 47516955, "title": "Filters in topology" }

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