Countably generated space

In mathematics, a topological space X {\displaystyle X} is called countably generated if the topology of X {\displaystyle 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.

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

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

A topological space X {\displaystyle X} has countable fan tightness if for every point x ∈ X {\displaystyle x\in X} and every sequence A 1 , A 2 , … {\displaystyle A_{1},A_{2},\ldots } of subsets of the space X {\displaystyle X} such that x ∈ ⋂ n A n ¯ = A 1 ¯ ∩ A 2 ¯ ∩ ⋯ , {\displaystyle x\in {\textstyle \bigcap \limits _{n}}\,{\overline {A_{n}}}={\overline {A_{1}}}\cap {\overline {A_{2}}}\cap \cdots ,} there are finite set B 1 ⊆ A 1 , B 2 ⊆ A 2 , … {\displaystyle B_{1}\subseteq A_{1},B_{2}\subseteq A_{2},\ldots } such that x ∈ ⋃ n B n ¯ = B 1 ∪ B 2 ∪ ⋯ ¯ . {\displaystyle x\in {\overline {{\textstyle \bigcup \limits _{n}}\,B_{n}}}={\overline {B_{1}\cup B_{2}\cup \cdots }}.}

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

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.

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.

• Finitely generated space – Type of topology in mathematicsPages displaying short descriptions of redirect targets
• Locally closed subset – Intersection of an open set and a closed set

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

• 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