Countably generated space

In mathematics, a topological space is called countably generated if the topology of 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 countably tight is used as well.

Definition

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A topological space   is called countably generated if for every subset     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

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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

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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

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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

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  • Finitely generated space – topology in which the intersection of any family of open sets is open
  • Locally closed subset
  • Tightness (topology) – Function that returns cardinal numbers − Tightness is a cardinal function related to countably generated spaces and their generalizations.

References

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