Relatively compact subset: Difference between revisions
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A subset of a [[topological space]] is termed '''relatively compact''' if its [[closure]] in the space is [[compact space|compact]]. | A subset of a [[topological space]] is termed '''relatively compact''' if its [[closure]] in the space is [[compact space|compact]]. | ||
Note that when the space is a [[Hausdorff space]], or more generally, a [[KC-space]], it suffices to say that the | Note that when the space is a [[Hausdorff space]], or more generally, a [[KC-space]], it suffices to say that the subset is contained in a compact subset. | ||
(The equivalence of definitions relies on the fact that [[compactness is weakly hereditary|a closed subset of a compact space is compact]] and [[Hausdorff implies KC|a compact subset of a Hausdorff space is closed]]). | |||
Revision as of 16:42, 22 January 2008
This article defines a property over pairs of a topological space and a subspace, or equivalently, properties over subspace embeddings (viz, subsets) in topological spaces
Definition
A subset of a topological space is termed relatively compact if its closure in the space is compact.
Note that when the space is a Hausdorff space, or more generally, a KC-space, it suffices to say that the subset is contained in a compact subset. (The equivalence of definitions relies on the fact that a closed subset of a compact space is compact and a compact subset of a Hausdorff space is closed).