Compact space

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This article defines a property of topological space that is pivotal (viz important) among currently studied properties of topological spaces

Definition

A topological space is said to be compact if it satisfies the following equivalent conditions:

Relation with other properties

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Weaker properties

Metaproperties

Products

This property of topological spaces is closed under taking arbitrary products
View all properties of topological spaces closed under products

Any product of compact spaces is compact. This result is true only in theproduct topology, not in the box topology. The result is known as the Tychonoff theorem. For the case of finite direct products, there is a much simpler proof that makes use of the tube lemma.

Subspaces

Any closed subset of a compact space is compact. In fact, given any Hausdorff space, every compact subset is closed, so we cannot in general hope for too many compact sets under than the closed ones.