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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.
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 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.
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.
See also
- The concept of finitely generated space is related to this notion.
- Tightness is a cardinal function related to countably generated spaces and their generalizations.
External links
- A Glossary of Definitions from General Topology [1]
- http://thales.doa.fmph.uniba.sk/density/pages/slides/sleziak/paper.pdf
References
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