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.
A topological space
is called countably generated if for every subset
whenever for each countable subspace
the set
is closed in
is countably generated if and only if the closure of any
equals the union of closures of all countable subsets of
A topological space
has countable fan tightness if for every point
and every sequence
of subsets of the space
there are finite set
A topological space
has countable strong fan tightness if for every point
of subsets of the space
Every strong Fréchet–Urysohn space has strong countable fan tightness.
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.
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.
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