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 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 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.