Countably generated space

Definition

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

Countable fan tightness

A topological space X has countable fan tightness if for every point x ∈ X and every sequence A 1 , A 2 , … of subsets of the space X such that x ∈ ⋂ n A n ¯ , there are finite set B 1 ⊂ A 1 , B 2 ⊂ A 2 , … such that x ∈ ⋃ n B n ¯ .

A topological space X has countable strong fan tightness if for every point x ∈ X and every sequence A 1 , A 2 , … of subsets of the space X such that x ∈ ⋂ n A n ¯ , there are points x 1 ⊂ A 1 , x 2 ⊂ A 2 , … such that x ∈ { x 1 , x 2 , … } ¯ . 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.