↑ arvau

Wikipedia / Countably generated space

In mathematics, a topological space is called countably generated 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 is used as well.

Definition

is called is coherent whenever for each countable subspace whenever for each countable subspace

is countably tight

Countable fan tightness

has

has 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

References