Countably generated space
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.
A topological space is called countably generated if 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.
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.
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.
- A Glossary of Definitions from General Topology
- Herrlich, Horst (1968). Topologische Reflexionen und Coreflexionen. Lecture Notes in Math. 78. Berlin: Springer.