Countably generated space

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

Definition

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

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

Countable fan tightness

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

A topological space ${\displaystyle X}$ has countable strong fan tightness if for every point ${\displaystyle x\in X}$ and every sequence ${\displaystyle A_{1},A_{2},\ldots }$ of subsets of the space ${\displaystyle X}$ such that ${\displaystyle x\in {\textstyle \bigcap \limits _{n}}\,{\overline {A_{n}}}={\overline {A_{1}}}\cap {\overline {A_{2}}}\cap \cdots ,}$ there are points ${\displaystyle x_{1}\in A_{1},x_{2}\in A_{2},\ldots }$ such that ${\displaystyle x\in {\overline {\left\{x_{1},x_{2},\ldots \right\}}}.}$ 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 – Type of topology in mathematicsPages displaying short descriptions of redirect targets
• Locally closed subset – Intersection of an open set and a closed set

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