Hereditarily compact space: Difference between revisions
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A [[topological space]] is termed '''hereditarily compact''' if every subset of it is [[compact space|compact]] in the [[subspace topology]]. Note that a hereditarily compact Hausdorff space must be discrete, so hereditarily compact spaces are not very common. | A [[topological space]] is termed '''hereditarily compact''' if every subset of it is [[compact space|compact]] in the [[subspace topology]]. Note that a hereditarily compact Hausdorff space must be discrete, so hereditarily compact spaces are not very common. | ||
==Relation with other properties== | |||
===Stronger properties=== | |||
* [[Noetherian space]] | |||
===Weaker properties=== | |||
* [[Compact space]] | |||
Revision as of 20:03, 13 January 2008
This article defines a property of topological spaces: a property that can be evaluated to true/false for any topological space|View a complete list of properties of topological spaces
Definition
Symbol-free definition
A topological space is termed hereditarily compact if every subset of it is compact in the subspace topology. Note that a hereditarily compact Hausdorff space must be discrete, so hereditarily compact spaces are not very common.