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Countably generated space

Wikipedia contributorsAdapted for Fathomly · View authors
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Source updated July 27, 2026

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.

01Definition

A topological space X is called countably generated if the topology on X is coherent with the family of its countable subspaces. In other words, any subset V\subseteq X is closed in X whenever for each countable subspace U of X the set V\cap U is closed in U; or equivalently, any subset V\subseteq X is open in X whenever for each countable subspace U of X the set V\cap U is open in U.

Equivalently, X is countably tight; that is, for every set A\subseteq X and every point x\in {\overline {A}}, there is a countable set D\subseteq A with x\in {\overline {D}}. In other words, the closure of A is the union of the closures of all countable subsets of A.

02Countable 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.

03Properties

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.

04Examples

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.

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Sources and credits

This article is adapted from the Wikipedia article “Countably generated space”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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