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In mathematics, a topological space $X$ is called countably generated if the topology of $X$ is determined by the countable sets in a similar way as the topology of a sequential space (or a Fréchet space) is determined by the convergent sequences.

The countably generated spaces are precisely the spaces having countable tightness—therefore the name countably tight is used as well.

Definition

A topological space $X$ is called countably generated if for every subset $V\subseteq X,$ $V$ is closed in $X$ whenever for each countable subspace $U$ of $X$ the set $V\cap U$ is closed in $U$ . Equivalently, $X$ is countably generated if and only if the closure of any $A\subseteq X$ equals the union of closures of all countable subsets of $A.$

Countable fan tightness

A topological space $X$ has countable fan tightness if for every point $x\in X$ and every sequence $A_{1},A_{2},\ldots$ of subsets of the space $X$ such that $x\in {\textstyle \bigcap \limits _{n}}\,{\overline {A_{n}}}={\overline {A_{1}}}\cap {\overline {A_{2}}}\cap \cdots ,$ there are finite set $B_{1}\subseteq A_{1},B_{2}\subseteq A_{2},\ldots$ such that $x\in {\overline {{\textstyle \bigcup \limits _{n}}\,B_{n}}}={\overline {B_{1}\cup B_{2}\cup \cdots }}.$

A topological space $X$ has countable strong fan tightness if for every point $x\in X$ and every sequence $A_{1},A_{2},\ldots$ of subsets of the space $X$ such that $x\in {\textstyle \bigcap \limits _{n}}\,{\overline {A_{n}}}={\overline {A_{1}}}\cap {\overline {A_{2}}}\cap \cdots ,$ there are points $x_{1}\in A_{1},x_{2}\in A_{2},\ldots$ such that $x\in {\overline {\left\{x_{1},x_{2},\ldots \right\}}}.$ Every strong Fréchet–Urysohn space has strong countable fan tightness.

Properties

A quotient of a countably generated space is again countably generated. Similarly, a topological sum of countably generated spaces is countably generated. Therefore, the countably generated spaces form a coreflective subcategory of the category of topological spaces. They are the coreflective hull of all countable spaces.

Any subspace of a countably generated space is again countably generated.

Examples

Every sequential space (in particular, every metrizable space) is countably generated.

An example of a space which is countably generated but not sequential can be obtained, for instance, as a subspace of Arens–Fort space.

References

Herrlich, Horst (1968). Topologische Reflexionen und Coreflexionen. Lecture Notes in Math. 78. Berlin: Springer.

External links

A Glossary of Definitions from General Topology [1]
https://web.archive.org/web/20040917084107/http://thales.doa.fmph.uniba.sk/density/pages/slides/sleziak/paper.pdf

This topology-related article is a stub. You can help Wikipedia by expanding it.

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-In [[mathematics]], a [[topological space]] ''X'' is called '''countably generated''' if the topology of ''X'' is determined by the [[countable]] sets in a similar way as the topology of a [[sequential space]] (or a [[Fréchet space]]) by the convergent sequences.
+In [[mathematics]], a [[topological space]] <math>X</math> is called '''countably generated''' if the topology of <math>X</math> is determined by the [[countable]] sets in a similar way as the topology of a [[sequential space]] (or a [[Fréchet space]]) is determined by the convergent sequences.
-The countable generated spaces are precisely the spaces having countable [[tightness (topology)|tightness]] - therefore the name ''countably tight'' is used as well.
+The countably generated spaces are precisely the spaces having countable [[Tightness (topology)|tightness]]—therefore the name '''{{visible anchor|countably tight}}''' is used as well.
 ==Definition==
-A topological space ''X'' is called '''countably generated''' if ''V'' is closed in ''X'' whenever for each countable [[Subspace (topology)|subspace]] ''U'' of ''X'' the set <math>V \cap U</math> is closed in ''U''. Equivalently, ''X'' is countably generated if and only if the closure of any subset ''A'' of ''X'' equals the union of closures of all countable subsets of ''A''.
+A topological space <math>X</math> is called '''{{visible anchor|countably generated}}''' if for every subset <math>V \subseteq X,</math> <math>V</math> is closed in <math>X</math> whenever for each countable [[Subspace (topology)|subspace]] <math>U</math> of <math>X</math> the set <math>V \cap U</math> is closed in <math>U</math>. Equivalently, <math>X</math> is countably generated if and only if the closure of any <math>A \subseteq X</math> equals the union of closures of all countable subsets of <math>A.</math>
+==Countable fan tightness==
-A [[quotient space|quotient]] of countably generated space is again countably generated. Similarly, a [[topological sum]] of countably generated spaces is countably generated. Therefore the countably generated spaces form a [[coreflective subcategory]] of the [[category of topological spaces]]. They are the coreflective hull of all countable spaces.
+A topological space <math>X</math> has '''{{visible anchor|countable fan tightness}}''' if for every point <math>x \in X</math> and every sequence <math>A_1, A_2, \ldots</math> of subsets of the space <math>X</math> such that <math>x \in {\textstyle\bigcap\limits_n} \, \overline{A_n} = \overline{A_1} \cap \overline{A_2} \cap \cdots,</math> there are finite set <math>B_1\subseteq A_1, B_2 \subseteq A_2, \ldots</math> such that <math>x \in \overline{{\textstyle\bigcup\limits_n} \, B_n} = \overline{B_1 \cup B_2 \cup \cdots}.</math>
+A topological space <math>X</math> has '''{{visible anchor|countable strong fan tightness}}''' if for every point <math>x \in X</math> and every sequence <math>A_1, A_2, \ldots</math> of subsets of the space <math>X</math> such that <math>x \in {\textstyle\bigcap\limits_n} \, \overline{A_n} = \overline{A_1} \cap \overline{A_2} \cap \cdots,</math> there are points <math>x_1 \in A_1, x_2 \in A_2, \ldots</math> such that <math>x \in \overline{\left\{x_1, x_2, \ldots\right\}}.</math> Every [[strong Fréchet–Urysohn space]] has strong countable fan tightness.
+==Properties==
+A [[Quotient space (topology)|quotient]] of a countably generated space is again countably generated. Similarly, a [[topological sum]] of countably generated spaces is countably generated. Therefore, the countably generated spaces form a [[coreflective subcategory]] of the [[category of topological spaces]]. They are the coreflective hull of all countable spaces.
 Any subspace of a countably generated space is again countably generated.
@@ Line 13: / Line 21: @@
 ==Examples==
-Every sequential space (in particular, every metrizable space) is countably generated.
+Every sequential space (in particular, every [[metrizable space]]) is countably generated.
 An example of a space which is countably generated but not sequential can be obtained, for instance, as a subspace of [[Arens–Fort space]].
@@ Line 19: / Line 27: @@
 ==See also==
-* The concept of [[finitely generated space]] is related to this notion.
+* {{annotated link|Finitely generated space}}
+* {{annotated link|Locally closed subset}}
-* [Tightness_(topology)|Tightness]
+* {{annotated link|Tightness (topology)}} − Tightness is a cardinal function related to countably generated spaces and their generalizations.
-==External links==
-* A Glossary of Definitions from General Topology [http://math.berkeley.edu/~apollo/topodefs.ps]
-* http://thales.doa.fmph.uniba.sk/density/pages/slides/sleziak/paper.pdf
 ==References==
-*{{cite book | last=Herrlich | first=Horst | title=Topologische Reflexionen und Coreflexionen | publisher = [[Springer Science+Business Media|Springer]]| location=Berlin | year=1968 | id= | others=Lecture Notes in Math. 78}}
+{{reflist}}
+{{reflist|group=note}}
+* {{cite book|last=Herrlich|first=Horst|authorlink=Horst Herrlich|title=Topologische Reflexionen und Coreflexionen|publisher=[[Springer Science+Business Media|Springer]]|location=Berlin|year=1968|others=Lecture Notes in Math. 78}}
+==External links==
+* A Glossary of Definitions from General Topology [https://web.archive.org/web/20120204044336/https://math.berkeley.edu/~apollo/topodefs.ps]
+* https://web.archive.org/web/20040917084107/http://thales.doa.fmph.uniba.sk/density/pages/slides/sleziak/paper.pdf
 [[Category:General topology]]