Compact Set

analysis topology

Definition

Compact Set

A set in a topological space is called compact if every open cover of has a finite subcover.

That is, if where each is an open set, then there exists a finite subset such that:

Heine-Borel Theorem

In the context of Euclidean space (), compactness is characterised by the following theorem:

Heine-Borel

A subset is compact if and only if it is both closed and bounded.

Properties

• Bolzano-Weierstrass: A set is compact if and only if every sequence in has a subsequence that converges to a point in (sequential compactness).
• Extreme Value Theorem: A continuous function on a compact set attains its maximum and minimum values.
• Uniform Continuity: A continuous function on a compact set is uniformly continuous.
• Image under Continuity: The image of a compact set under a continuous mapping is compact.