Compact Set

analysis topology

Definition

Compact Set

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

That is, if $K\subseteq {\bigcup }_{i\in I}{U}_i$ where each $U_i$ is an open set, then there exists a finite subset $\{i_1,i_2,\dots ,i_n\}\subseteq I$ such that:

$$K\subseteq U\ast i_1\cup U\ast i\ast 2\cup \cdots \cup U\ast i_n$$

Heine-Borel Theorem

In the context of Euclidean space (${\mathbb{R}}^n$), compactness is characterised by the following theorem:

Heine-Borel

A subset $K\subseteq {\mathbb{R}}^n$ is compact if and only if it is both closed and bounded.

Properties

• Bolzano-Weierstrass: A set $K$ is compact if and only if every sequence in $K$ has a subsequence that converges to a point in $K$ (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.