{{Short description|Rule in mathematics}} {{distinguish|selection bias|anthropic principle}} [[File:S1ABanimated.gif|thumb|An illustration of the selection principle <math>\text{S}_1(\mathbf{A},\mathbf{B})</math>]] In mathematics, a '''selection principle''' is a rule asserting the possibility of obtaining mathematically significant objects by selecting elements from given [[sequence]]s of [[Set (mathematics)|set]]s. The theory of '''selection principles''' studies these principles and their relations to other mathematical properties. Selection principles mainly describe covering properties, [[measure theory|measure-]] and [[category-theoretic]] properties, and local properties in [[topological space]]s, especially [[function space]]s. Often, the characterization of a mathematical property using a selection principle is a nontrivial task leading to new insights on the characterized property.
==The main selection principles== In 1924, [[Karl Menger]] <ref name="Karl Menger">{{cite journal|last1=Menger|first1=Karl|title=Einige Überdeckungssätze der Punktmengenlehre| journal=Sitzungsberichte der Wiener Akademie|date=1924|volume=133| pages=421–444|jfm=50.0129.01}} Reprinted in ''Selecta Mathematica I'' (2002), {{doi|10.1007/978-3-7091-6110-4_14}}, {{isbn|978-3-7091-7282-7}}, pp. 155-178.</ref> introduced the following basis property for [[metric space]]s: Every [[basis (topology)|basis]] of the topology contains a sequence of sets with vanishing diameters that covers the space. Soon thereafter, [[Witold Hurewicz]]<ref name=":0">{{cite journal|last1=Hurewicz|first1=Witold| title=Über eine verallgemeinerung des Borelschen Theorems| journal=[[Mathematische Zeitschrift]]|date=1926|volume=24|issue=1|pages=401–421|doi=10.1007/bf01216792|s2cid=119867793}}</ref> observed that Menger's basis property is equivalent to the following selective property: for every sequence of [[open cover]]s of the space, one can select finitely many open sets from each cover in the sequence, such that the family of all selected sets covers the space. Topological spaces having this covering property are called '''Menger spaces'''.
Hurewicz's reformulation of Menger's property was the first important [[topological property]] described by a selection principle. Let <math>\mathbf{A}</math> and <math>\mathbf{B}</math> be classes of mathematical objects. In 1996, [[Marion Scheepers]]<ref name=coc1>{{cite journal|last1=Scheepers|first1=Marion| title=Combinatorics of open covers I: Ramsey theory| journal=[[Topology and Its Applications]]|date=1996|volume=69|pages=31–62|doi=10.1016/0166-8641(95)00067-4|doi-access=free}}</ref> introduced the following selection hypotheses, capturing a large number of classic mathematical properties:
* <math>\text{S}_1(\mathbf{A},\mathbf{B})</math>''':''' For every sequence <math>\mathcal{U}_1,\mathcal{U}_2,\ldots</math> of elements from the class <math>\mathbf{A}</math>, there are elements <math>U_1\in\mathcal{U}_1,U_2\in\mathcal{U}_2,\dots</math> such that <math>\{U_n:n\in\mathbb{N}\}\in\mathbf{B}</math>. * <math>\text{S}_{\text{fin}}(\mathbf{A},\mathbf{B})</math>''':''' For every sequence <math>\mathcal{U}_1,\mathcal{U}_2,\ldots</math> of elements from the class <math>\mathbf{A}</math>, there are finite subsets <math>\mathcal{F}_1\subseteq\mathcal{U}_1,\mathcal{F}_2\subseteq\mathcal{U}_2,\dots</math> such that <math>\bigcup_{n=1}^\infty \mathcal{F}_n\in\mathbf{B}</math>.
In the case where the classes <math>\mathbf{A}</math> and <math>\mathbf{B}</math> consist of covers of some ambient space, Scheepers also introduced the following selection principle.
* <math>\text{U}_{\text{fin}}(\mathbf{A},\mathbf{B})</math>''':''' For every sequence <math>\mathcal{U}_1,\mathcal{U}_2,\ldots</math> of elements from the class <math>\mathbf{A}</math>, none containing a finite subcover, there are finite subsets <math>\mathcal{F}_1\subseteq\mathcal{U}_1,\mathcal{F}_2\subseteq\mathcal{U}_2,\dots</math> such that <math>\{\bigcup \mathcal{F}_1, \bigcup \mathcal{F}_2,\dotsc\}\in\mathbf{B}</math>.
Later, [[Boaz Tsaban]] identified the prevalence of the following related principle: *<math>\binom{\mathbf{A}}{\mathbf{B}}</math>: Every member of the class <math>\mathbf{A}</math> includes a member of the class <math>\mathbf{B}</math>.
The notions thus defined are ''selection principles''. An instantiation of a selection principle, by considering specific classes <math>\mathbf{A}</math> and <math>\mathbf{B}</math>, gives a ''selection (or: selective) property''. However, these terminologies are used interchangeably in the literature.
===Variations===
For a set <math>A\subset X</math> and a family <math>\mathcal{F}</math> of subsets of <math>X</math>, the '''star of <math>A</math> in <math>\mathcal{F}</math>''' is the set <math>\text{St}(A,\mathcal{F})=\bigcup\{F\in\mathcal{F}:A\cap F\neq\emptyset\}</math>.
In 1999, [[Ljubisa D.R. Kocinac]] introduced the following ''star selection principles'':<ref name=Kssp>{{cite journal|last1=Kocinac|first1=Ljubisa D. R.| title=Star selection principles: a survey| journal=Khayyam Journal of Mathematics|date=2015|volume=1|pages=82–106| url=http://emis.de/journals/KJM/}}</ref>
* <math>\text{S}_1^*(\mathbf{A},\mathbf{B})</math>''':''' For every sequence <math>\mathcal{U}_1,\mathcal{U}_2,\ldots</math> of elements from the class <math>\mathbf{A}</math>, there are elements <math>U_1\in\mathcal{U}_1,U_2\in\mathcal{U}_2,\dots</math> such that <math>\{\text{St}(U_n,\mathcal{U}_n):n\in\mathbb{N}\}\in\mathbf{B}</math>. * <math>\text{S}_{\text{fin}}^*(\mathbf{A},\mathbf{B})</math>''':''' For every sequence <math>\mathcal{U}_1,\mathcal{U}_2,\ldots</math> of elements from the class <math>\mathbf{A}</math>, there are finite subsets <math>\mathcal{F}_1\subseteq\mathcal{U}_1,\mathcal{F}_2\subseteq\mathcal{U}_2,\dots</math> such that <math>\{\text{St}(\bigcup \mathcal{F}_n,\mathcal{U}_n):n\in\mathbb{N}\}\in\mathbf{B}</math>.
The star selection principles are special cases of the general selection principles. This can be seen by modifying the definition of the family <math>\mathbf{B}</math> accordingly.
==Covering properties==
Covering properties form the kernel of the theory of selection principles. Selection properties that are not covering properties are often studied by using implications to and from selective covering properties of related spaces.
Let <math>X</math> be a [[topological space]]. An ''open cover'' of <math>X</math> is a family of open sets whose union is the entire space <math>X.</math> For technical reasons, we also request that the entire space <math>X</math> is not a member of the cover. The class of open covers of the space <math>X</math> is denoted by <math>\mathbf{O}</math>. (Formally, <math>\mathbf{O}(X)</math>, but usually the space <math>X</math> is fixed in the background.) The above-mentioned property of Menger is, thus, <math>\text{S}_{\text{fin}}(\mathbf{O},\mathbf{O})</math>. In 1942, Fritz Rothberger considered Borel's strong measure zero sets, and introduced a topological variation later called [[Rothberger space]] (also known as ''C<math>''</math> space''). In the notation of selections, Rothberger's property is the property <math>\text{S}_{1}(\mathbf{O},\mathbf{O})</math>.
An open cover <math>\mathcal{U}</math> of <math>X</math> is '''point-cofinite''' if it has infinitely many elements, and every point <math>x\in X</math> belongs to all but finitely many sets <math>U\in\mathcal{U}</math>. (This type of cover was considered by Gerlits and Nagy, in the third item of a certain list in their paper. The list was enumerated by Greek letters, and thus these covers are often called '''<math>\gamma</math>-covers'''.) The class of point-cofinite open covers of <math>X</math> is denoted by <math>\mathbf{\Gamma}</math>. A topological space is a [[Hurewicz space]] if it satisfies <math>\text{U}_{\text{fin}}(\mathbf{O},\mathbf{\Gamma})</math>.
An open cover <math>\mathcal{U}</math> of <math>X</math> is an '''<math>\omega</math>-cover''' if every finite subset of <math>X</math> is contained in some member of <math>\mathcal{U}</math>. The class of <math>\omega</math>-covers of <math>X</math> is denoted by <math>\mathbf{\Omega}</math>. A topological space is a [[γ-space]] if it satisfies <math>\binom{\mathbf{\Omega}}{\mathbf{\Gamma}}</math>.
By using star selection hypotheses one obtains properties such as '''star-Menger''' (<math>\text{S}_{\text{fin}}^*(\mathbf{O},\mathbf{O})</math>), '''star-Rothberger''' (<math>\text{S}_1^*(\mathbf{O},\mathbf{O})</math>) and '''star-Hurewicz''' (<math>\text{S}_{\text{fin}}^*(\mathbf{O},\mathbf{\Gamma})</math>).
===The Scheepers diagram=== There are 36 selection properties of the form <math> \Pi(\mathbf{A},\mathbf{B})</math>, for <math>\Pi\in\{\text{S}_1, \text{S}_\text{fin},\text{U}_\text{fin}, \bigl(~~\bigr)\}</math> and <math>\mathbf{A},\mathbf{B}\in\{\mathbf{O},\mathbf{\Gamma},\mathbf{\Omega}\}</math>. Some of them are trivial (hold for all spaces, or fail for all spaces). Restricting attention to [[Lindelöf space]]s, the diagram below, known as the ''Scheepers diagram'',<ref name=coc1 /><ref name="coc2">{{cite journal|last2=Miller|first2=Arnold|last3=Scheepers|first3=Marion|last4=Szeptycki|first4=Paul|date=1996|title=Combinatorics of open covers II|journal=Topology and Its Applications|volume=73|issue=3|pages=241–266|doi=10.1016/S0166-8641(96)00075-2|last1=Just|first1=Winfried|arxiv=math/9509211|s2cid=14946860}}</ref> presents nontrivial selection properties of the above form, and every nontrivial selection property is equivalent to one in the diagram. Arrows denote implications.
{| style="margin: 1em auto 1em auto;" |- | [[File:ScheepersDiagram.jpg|The Scheepers Diagram|847x847px]] |}
==Local properties==
Selection principles also capture important local properties.
Let <math>Y</math> be a topological space, and <math>y\in Y</math>. The class of sets <math>A</math> in the space <math>Y</math> that have the point <math>y</math> in their closure is denoted by <math>\mathbf{\Omega_y}</math>. The class <math>\mathbf{\Omega^{\text{ctbl}}_y}</math> consists of the ''countable'' elements of the class <math>\mathbf{\Omega_y}</math>. The class of sequences in <math>Y</math> that converge to <math>y</math> is denoted by <math>\mathbf{\Gamma_y}</math>.
* A space <math>Y</math> is [[Fréchet–Urysohn space|Fréchet–Urysohn]] [[if and only if]] it satisfies <math>\binom{\mathbf{\Omega_y}}{\mathbf{\Gamma_y}}</math> for all points <math>y\in Y</math>. * A space <math>Y</math> is [[Fréchet–Urysohn space|strongly Fréchet–Urysohn]] if and only if it satisfies <math>\text{S}_1(\mathbf{\Omega_y},\mathbf{\Gamma_y})</math> for all points <math>y\in Y</math>. * A space <math>Y</math> has [[countable tightness]] if and only if it satisfies <math>\binom{\mathbf{\Omega_y}}{\mathbf{\Omega^{\text{ctbl}}_y}}</math> for all points <math>y\in Y</math>. * A space <math>Y</math> has [[countable tightness|countable fan tightness]] if and only if it satisfies <math>\text{S}_{\text{fin}}(\mathbf{\Omega_y},\mathbf{\Omega_y})</math> for all points <math>y\in Y</math>. * A space <math>Y</math> has [[countable tightness|countable strong fan tightness]] if and only if it satisfies <math>\text{S}_{1}(\mathbf{\Omega_y},\mathbf{\Omega_y})</math> for all points <math>y\in Y</math>.
==Topological games==
There are close connections between selection principles and [[topological game]]s.
===The Menger game===
Let <math>X</math> be a topological space. The Menger game <math>\text{G}_{\text{fin}}(\mathbf{O},\mathbf{O})</math> played on <math>X</math> is a game for two players, Alice and Bob. It has an inning per each natural number <math>n</math>. At the <math>n^{th}</math> inning, Alice chooses an open cover <math>\mathcal{U}_n</math> of <math>X</math>, and Bob chooses a finite subset <math>\mathcal{F}_n</math> of <math>\mathcal{U}</math>. If the family <math>\bigcup_{n=1}^\infty \mathcal{F}_n</math> is a cover of the space <math>X</math>, then Bob wins the game. Otherwise, Alice wins.
A '''strategy''' for a player is a function determining the move of the player, given the earlier moves of both players. A strategy for a player is a '''winning strategy''' if each play where this player sticks to this strategy is won by this player.
* A topological space is <math>\text{S}_{\text{fin}}(\mathbf{O},\mathbf{O})</math> if and only if Alice has no winning strategy in the game <math>\text{G}_{\text{fin}}(\mathbf{O},\mathbf{O})</math> played on this space.<ref name=":0" /><ref name="coc1" /> * Let <math>X</math> be a metric space. Bob has a winning strategy in the game <math>\text{G}_{\text{fin}}(\mathbf{O},\mathbf{O})</math> played on the space <math>X</math> if and only if the space <math>X</math> is <math>\sigma</math>-compact.<ref>{{Cite journal|last=Scheepers|first=Marion|date=1995-01-01|title=A direct proof of a theorem of Telgársky|journal=Proceedings of the American Mathematical Society|volume=123|issue=11|pages=3483–3485|doi=10.1090/S0002-9939-1995-1273523-1|issn=0002-9939|doi-access=free}}</ref><ref>{{Cite journal|last=Telgársky|first=Rastislav|date=1984-06-01|title=On games of Topsoe.|journal=Mathematica Scandinavica|volume=54|pages=170–176|doi=10.7146/math.scand.a-12050|issn=1903-1807|doi-access=free}}</ref>
Note that among Lindelöf spaces, metrizable is equivalent to regular and second-countable, and so the previous result may alternatively be obtained by considering '''limited information strategies'''.<ref>{{cite journal | last=Steven | first= Clontz | title=Applications of limited information strategies in Menger's game | journal=Commentationes Mathematicae Universitatis Carolinae | publisher=Charles University in Prague, Karolinum Press | volume=58 | issue=2 | date=2017-07-31 | issn=0010-2628 | doi=10.14712/1213-7243.2015.201 | pages=225–239 | doi-access=free }}</ref> A '''Markov''' strategy is one that only uses the most recent move of the opponent and the current round number.
* Let <math>X</math> be a regular space. Bob has a winning Markov strategy in the game <math>\text{G}_{\text{fin}}(\mathbf{O},\mathbf{O})</math> played on the space <math>X</math> if and only if the space <math>X</math> is <math>\sigma</math>-compact. * Let <math>X</math> be a [[second-countable space]]. Bob has a winning Markov strategy in the game <math>\text{G}_{\text{fin}}(\mathbf{O},\mathbf{O})</math> played on the space <math>X</math> if and only if he has a winning perfect-information strategy.
In a similar way, we define games for other selection principles from the given Scheepers Diagram. In all these cases a topological space has a property from the Scheepers Diagram if and only if Alice has no winning strategy in the corresponding game.<ref name="pawlikowski">{{Cite journal|last=Pawlikowski|first=Janusz|title=Undetermined sets of point-open games|url=https://eudml.org/doc/212029|journal=Fundamenta Mathematicae|volume=144|issue=3|pages=279–285|issn=0016-2736|year=1994|doi=10.4064/fm-144-3-279-285 }}</ref> But this does not hold in general: Let <math>\mathbf{K}</math> be the family of k-covers of a space. That is, such that every compact set in the space is covered by some member of the cover. Francis Jordan demonstrated a space where the selection principle <math>\text{S}_1(\mathbf{K},\mathbf{O})</math> holds, but Alice ''has'' a winning strategy for the game <math>\text{G}_1(\mathbf{K},\mathbf{O})</math> <ref>{{cite journal | last=Jordan | first=Francis | title=On the instability of a topological game related to consonance | journal=Topology and Its Applications | publisher=Elsevier BV | volume=271 | year=2020 | issn=0166-8641 | doi=10.1016/j.topol.2019.106990 | article-number=106990 | s2cid=213386675 | doi-access=free }}</ref>
==Examples and properties== * Every <math>\text{S}_{\text{fin}}(\mathbf{O},\mathbf{O})</math> space is a [[Lindelöf space]]. * Every [[σ-compact space]] (a countable union of compact spaces) is <math>\text{U}_{\text{fin}}(\mathbf{O},\mathbf{\Gamma})</math>. * <math>\binom{\mathbf{\Omega}}{\mathbf{\Gamma}}\Rightarrow\text{U}_{\text{fin}}(\mathbf{O},\mathbf{\Gamma})\Rightarrow\text{S}_{\text{fin}}(\mathbf{O},\mathbf{O})</math>. * <math>\binom{\mathbf{\Omega}}{\mathbf{\Gamma}}\Rightarrow\text{S}_{1}(\mathbf{O},\mathbf{O})\Rightarrow\text{S}_{\text{fin}}(\mathbf{O},\mathbf{O})</math>. * Assuming the [[Continuum Hypothesis]], there are sets of real numbers witnessing that the above implications cannot be reversed.<ref name="coc2" /> * Every [[Luzin space|Luzin set]] is <math>\text{S}_{\text{fin}}(\mathbf{O},\mathbf{O})</math> but no <math>\text{U}_{\text{fin}}(\mathbf{O},\mathbf{\Gamma})</math>.<ref name=":1">{{Cite journal|last=Rothberger|first=Fritz|year=1938|title=Eine Verschärfung der Eigenschaft C|url=http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.bwnjournal-article-fmv30i1p8bwm?q=bwmeta1.element.bwnjournal-number-fm-1938-30-1;7&qt=CHILDREN-STATELESS|journal=Fundamenta Mathematicae|volume=30|pages=50–55|doi=10.4064/fm-30-1-50-55|doi-access=free}}</ref><ref>{{Cite journal|last=Hurewicz|first=Witold|year=1927|title=Über Folgen stetiger Funktionen|url=http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.bwnjournal-article-fmv9i1p17bwm?q=bwmeta1.element.bwnjournal-number-fm-1927-9-1;16&qt=CHILDREN-STATELESS|journal=Fundamenta Mathematicae|volume=9|pages=193–210|doi=10.4064/fm-9-1-193-210|doi-access=free}}</ref> * Every [[Sierpiński set]] is Hurewicz.<ref>{{cite journal|last1=Fremlin|first1=David|last2=Miller|first2=Arnold| title=On some properties of Hurewicz, Menger and Rothberger| journal=Fundamenta Mathematicae|date=1988|volume=129|pages=17–33| url=https://matwbn.icm.edu.pl/ksiazki/fm/fm129/fm12913.pdf|doi=10.4064/fm-129-1-17-33|doi-access=free}}</ref>
Subsets of the real line <math>\mathbb{R}</math> (with the induced [[subspace topology]]) holding selection principle properties, most notably Menger and Hurewicz spaces, can be characterized by their continuous images in the [[Baire space (set theory)|Baire space]] <math>\mathbb{N}^\mathbb{N}</math>. For functions <math>f,g\in \mathbb{N}^\mathbb{N}</math>, write <math>f\leq^* g</math> if <math> f(n)\leq g(n)</math> for all but finitely many natural numbers <math> n</math>. Let <math>A</math> be a subset of <math>\mathbb{N}^\mathbb{N}</math>. The set <math> A </math> is '''bounded''' if there is a function <math> g\in\mathbb{N}^\mathbb{N}</math> such that <math> f\leq^* g</math> for all functions <math>f\in A</math>. The set <math> A </math> is '''dominating''' if for each function <math> f\in\mathbb{N}^\mathbb{N}</math> there is a function <math>g\in A</math> such that <math> f\leq^* g</math>.
* A subset of the real line is <math>\text{S}_{\text{fin}}(\mathbf{O},\mathbf{O})</math> if and only if every continuous image of that space into the Baire space is not dominating.<ref name=":2">{{Cite journal|last=Recław|first=Ireneusz|year=1994|title=Every Lusin set is undetermined in the point-open game|url=http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.bwnjournal-article-fmv144i1p43bwm?q=bwmeta1.element.bwnjournal-number-fm-1994-144-1;3&qt=CHILDREN-STATELESS|journal=Fundamenta Mathematicae|volume=144|pages=43–54|doi=10.4064/fm-144-1-43-54|doi-access=free}}</ref> * A subset of the real line is <math>\text{U}_{\text{fin}}(\mathbf{O},\mathbf{\Gamma})</math> if and only if every continuous image of that space into the Baire space is bounded.<ref name=":2" />
==Connections with other fields==
===General topology===
* Every <math>\text{S}_{\text{fin}}(\mathbf{O},\mathbf{O})</math> space is a [[D-space]].<ref>{{cite journal|last1=Aurichi|first1=Leandro|title=D-Spaces, Topological Games, and Selection Principles|journal=Topology Proceedings|date=2010|volume=36|pages=107–122|url=http://topology.auburn.edu/tp/reprints/v36/tp36009.pdf|archive-date=2017-01-09|access-date=2017-01-08|archive-url=https://web.archive.org/web/20170109113902/http://topology.auburn.edu/tp/reprints/v36/tp36009.pdf|url-status=dead}}</ref> Let '''P''' be a property of spaces. A space <math>X</math> is '''productively P''' if, for each space <math>Y</math> with property '''P''', the product space <math>X\times Y</math> has property '''P'''.
* Every [[separable space|separable]] productively [[paracompact]] space is <math>\text{U}_{\text{fin}}(\mathbf{O},\mathbf{\Gamma})</math>. * Assuming the [[Continuum Hypothesis]], every productively Lindelöf space is productively <math>\text{U}_{\text{fin}}(\mathbf{O},\mathbf{\Gamma})</math><ref>{{cite arXiv|last1=Szewczak|first1=Piotr|last2=Tsaban|first2=Boaz|title=Product of Menger spaces, II: general spaces|eprint=1607.01687|class=math.GN|year=2016}}</ref> * Let <math>A</math> be a <math>\binom{\mathbf{\Omega}}{\mathbf{\Gamma}}</math> subset of the real line, and <math>M</math> be a [[meagre set|meager]] subset of the real line. Then the set <math>A+M=\{a+x:a\in A, x\in M\}</math> is meager.<ref>{{cite journal|last1=Galvin|first1=Fred|last2=Miller|first2=Arnold|title=<math>\gamma</math>-sets and other singular sets of real numbers| journal=Topology and Its Applications|date=1984|volume=17|issue=2|pages=145–155|doi=10.1016/0166-8641(84)90038-5|doi-access=free}}</ref>
===Measure theory===
* Every <math>\text{S}_{1}(\mathbf{O},\mathbf{O})</math> subset of the real line is a [[strong measure zero set]].<ref name=":1" />
===Function spaces===
Let <math>X</math> be a [[Tychonoff space]], and <math>C(X)</math> be the space of continuous functions <math>f\colon X\to\mathbb{R}</math> with [[pointwise convergence]] topology. * <math>X</math> satisfies <math>\binom{\mathbf{\Omega}}{\mathbf{\Gamma}}</math> if and only if <math>C(X)</math> is [[Fréchet–Urysohn space|Fréchet–Urysohn]] if and only if <math>C(X)</math> is [[Fréchet–Urysohn space|strong Fréchet–Urysohn]].<ref>{{cite journal|last1=Gerlits|first1=J.|last2=Nagy|first2=Zs.|title=Some properties of <math>C(X)</math>, I| journal=Topology and Its Applications|date=1982|volume=14|issue=2|pages=151–161|doi=10.1016/0166-8641(82)90065-7|doi-access=free}}</ref> * <math>X</math> satisfies <math>\text{S}_{1}(\mathbf{\Omega},\mathbf{\Omega})</math> if and only if <math>C(X)</math> has [[countable tightness|countable strong fan tightness]].<ref>{{cite journal|last1=Sakai|first1=Masami|title=Property <math>C''</math> and function spaces| journal=[[Proceedings of the American Mathematical Society]]|date=1988|volume=104|issue=9|pages=917–919|doi=10.1090/S0002-9939-97-03897-5|doi-access=free}}</ref> * <math>X</math> satisfies <math>\text{S}_{\text{fin}}(\mathbf{\Omega},\mathbf{\Omega})</math> if and only if <math>C(X)</math> has [[countable tightness|countable fan tightness]].<ref>{{cite journal|date=1986|title=Hurewicz spaces, analytic sets and fan-tightness of spaces of functions|journal=[[Soviet Math. Dokl.]]|volume=2|pages=396–399|last1=Arhangel'skii|first1=Alexander|author-link=Alexander Arhangelskii}}</ref><ref name="coc2" />
== See also == * [[Compact space]] * [[Sigma-compact]] * [[Menger space]] * [[Hurewicz space]] * [[Rothberger space]]
==References== {{reflist}}
<!-- ==External links== --> {{Topology}}
[[Category:Properties of topological spaces]] [[Category:Topology]]