The countably generated spaces are precisely the spaces having countable
tightness—therefore the name countably tight is used as well.
Definition
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
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
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
Finitely generated space – topological space in which the intersection of any family of open sets is openPages displaying wikidata descriptions as a fallback
Tightness (topology) – Function that returns cardinal numbersPages displaying short descriptions of redirect targets − Tightness is a cardinal function related to countably generated spaces and their generalizations.
References
Herrlich, Horst (1968). Topologische Reflexionen und Coreflexionen. Lecture Notes in Math. 78. Berlin:
Springer.
External links
A Glossary of Definitions from General Topology
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