Countably generated space: Difference between revisions
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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''. |
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''. |
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==Properties== |
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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. |
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. |
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Revision as of 06:48, 9 July 2013
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
- Herrlich, Horst (1968). Topologische Reflexionen und Coreflexionen. Lecture Notes in Math. 78. Berlin: Springer.
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