en>Happysailor: rmv tag - tagged without discussion

2010-07-15T22:21:06Z

rmv tag - tagged without discussion

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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 (topology)|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 (topology)|subspace]] ''U'' of ''X'' the set <math>V \cap U</math> 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 space|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_(topology)|Tightness]] is a cardinal function related to countably generated spaces and their generalizations.

==External links==
* A Glossary of Definitions from General Topology [http://math.berkeley.edu/~apollo/topodefs.ps]
* http://thales.doa.fmph.uniba.sk/density/pages/slides/sleziak/paper.pdf

==References==
*{{cite book | last=Herrlich | first=Horst | title=Topologische Reflexionen und Coreflexionen | publisher = [[Springer Science+Business Media|Springer]]| location=Berlin | year=1968 | id= | others=Lecture Notes in Math. 78}}

[[Category:General topology]]

{{topology-stub}}

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en>Happysailor: rmv tag - tagged without discussion