# 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) by the convergent sequences.

The countable generated spaces are precisely the spaces having countable tightness - therefore the name *countably tight* is used as well.

## Definition

A topological space is called **countably generated** if is closed in whenever for each countable subspace of the set is closed in . Equivalently, is countably generated if and only if the closure of any equals the union of closures of all countable subsets of .

## Countable fan tightness

A topological space has **countable fan tightness** if for every point and every sequence of subsets of the space such that , there are finite set such that .

A topological space has **countable strong fan tightness** if for every point and every sequence of subsets of the space such that , there are points such that . Every strong Fréchet–Urysohn space has strong countable fan tightness.

## Properties

A quotient of 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

- The concept of finitely generated space is related to this notion.
- Tightness is a cardinal function related to countably generated spaces and their generalizations.

## External links

- A Glossary of Definitions from General Topology
- https://web.archive.org/web/20040917084107/http://thales.doa.fmph.uniba.sk/density/pages/slides/sleziak/paper.pdf

## References

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