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 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 $x\in \bar{A}$, there is a countable set $D\subseteq A$ with $x\in \bar{D}.$ In other words, the closure of $A$ is the union of the 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 $A_1,A_2,\dots$ of subsets of the space $X$ such that $x\in \bigcap _n\overline{A_n}=\overline{A_1}\cap \overline{A_2}\cap \cdots ,$ there are finite set $B_1\subseteq A_1,B_2\subseteq A_2,\dots$ such that $x\in \overline{\bigcup _n{B}_n}=\overline{B_1\cup B_2\cup \cdots }.$

A topological space $X$ has countable strong fan tightness if for every point $x\in X$ and every sequence $A_1,A_2,\dots$ of subsets of the space $X$ such that $x\in \bigcap _n\overline{A_n}=\overline{A_1}\cap \overline{A_2}\cap \cdots ,$ there are points $x_1\in A_1,x_2\in A_2,\dots$ such that $x\in \overline{\{x_1,x_2,\dots \}}.$ 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

• 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