Hereditarily compact space

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.

Relation with other properties

Stronger properties

• Noetherian space

Weaker properties

• Compact space
• Hereditarily paracompact space