Countably generated space
In mathematics, a topological space [math]\displaystyle{ X }[/math] is called countably generated if the topology of [math]\displaystyle{ X }[/math] 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
A topological space [math]\displaystyle{ X }[/math] is called countably generated if for every subset [math]\displaystyle{ V \subseteq X, }[/math] [math]\displaystyle{ V }[/math] is closed in [math]\displaystyle{ X }[/math] whenever for each countable subspace [math]\displaystyle{ U }[/math] of [math]\displaystyle{ X }[/math] the set [math]\displaystyle{ V \cap U }[/math] is closed in [math]\displaystyle{ U }[/math]. Equivalently, [math]\displaystyle{ X }[/math] is countably generated if and only if the closure of any [math]\displaystyle{ A \subseteq X }[/math] equals the union of closures of all countable subsets of [math]\displaystyle{ A. }[/math]
Countable fan tightness
A topological space [math]\displaystyle{ X }[/math] has countable fan tightness if for every point [math]\displaystyle{ x \in X }[/math] and every sequence [math]\displaystyle{ A_1, A_2, \ldots }[/math] of subsets of the space [math]\displaystyle{ X }[/math] such that [math]\displaystyle{ x \in {\textstyle\bigcap\limits_n} \, \overline{A_n} = \overline{A_1} \cap \overline{A_2} \cap \cdots, }[/math] there are finite set [math]\displaystyle{ B_1\subseteq A_1, B_2 \subseteq A_2, \ldots }[/math] such that [math]\displaystyle{ x \in \overline{{\textstyle\bigcup\limits_n} \, B_n} = \overline{B_1 \cup B_2 \cup \cdots}. }[/math]
A topological space [math]\displaystyle{ X }[/math] has countable strong fan tightness if for every point [math]\displaystyle{ x \in X }[/math] and every sequence [math]\displaystyle{ A_1, A_2, \ldots }[/math] of subsets of the space [math]\displaystyle{ X }[/math] such that [math]\displaystyle{ x \in {\textstyle\bigcap\limits_n} \, \overline{A_n} = \overline{A_1} \cap \overline{A_2} \cap \cdots, }[/math] there are points [math]\displaystyle{ x_1 \in A_1, x_2 \in A_2, \ldots }[/math] such that [math]\displaystyle{ x \in \overline{\left\{x_1, x_2, \ldots\right\}}. }[/math] 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
References
- Herrlich, Horst (1968). Topologische Reflexionen und Coreflexionen. Lecture Notes in Math. 78. Berlin: Springer.
External links
- A Glossary of Definitions from General Topology [1]
- https://web.archive.org/web/20040917084107/http://thales.doa.fmph.uniba.sk/density/pages/slides/sleziak/paper.pdf
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