Countably generated space

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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.

1 Definition
2 Examples
3 See also
4 External links

[edit] 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 $V \cap U$ 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.

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.