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# 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 for every subset ${\displaystyle V\subseteq X,}$ ${\displaystyle V}$ is closed in ${\displaystyle X}$ whenever for each countable subspace ${\displaystyle U}$ of ${\displaystyle X}$ the set ${\displaystyle V\cap U}$ is closed in ${\displaystyle U}$. Equivalently, ${\displaystyle X}$ is countably generated if and only if the closure of any ${\displaystyle A\subseteq X}$ equals the union of 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 ${\displaystyle x\in \bigcap _{n}{\overline {A_{n}}},}$ there are finite set ${\displaystyle B_{1}\subseteq A_{1},B_{2}\subseteq A_{2},\ldots }$ such that ${\displaystyle x\in {\overline {\bigcup _{n}B_{n}}}.}$

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 ${\displaystyle x\in \bigcap _{n}{\overline {A_{n}}},}$ there are points ${\displaystyle x_{1}\subseteq A_{1},x_{2}\subseteq A_{2},\ldots }$ 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.