PlanetMath: locally closed

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locally closed

(Definition)

Definition - A subset of a topological space is said to be locally closed if it is the intersection of an open and a closed subset.

The following result provides some equivalent definitions:

Proposition - The following are equivalent:

is locally closed in .
Each point in has an open neighborhood $U \subseteq X$ such that $U \cap Y$ is closed in (with the subspace topology).
is open in its closure $\overline{Y}$ (with the subspace topology).

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Cross-references: closure, subspace topology, closed, neighborhood, point, the following are equivalent, definitions, closed subset, open, intersection, topological space, subset
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This is version 2 of locally closed, born on 2007-10-25, modified 2007-10-25.
Object id is 10018, canonical name is LocallyClosed.
Accessed 851 times total.

Classification:

54D99 (General topology :: Fairly general properties :: Miscellaneous)

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