Countably generated space

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.

See also

• Finitely generated space – topological space in which the intersection of any family of open sets is openPages displaying wikidata descriptions as a fallback
• Locally closed subset
• Tightness (topology) – Function that returns cardinal numbersPages displaying short descriptions of redirect targets − Tightness is a cardinal function related to countably generated spaces and their generalizations.

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