Compactness

Definition

A topological space $X$ is compact if every open cover has a finite subcover.

Formally: if $\{U_{\alpha }{\}}_{\alpha \in A}$ is a collection of open sets with $X={\bigcup }_{\alpha }{U}_{\alpha }$, then there exist ${\alpha }_1,\dots ,{\alpha }_n$ such that:

$X=U_{{\alpha }_1}\cup \cdots \cup U_{{\alpha }_n}$

Key Theorems

Heine-Borel

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

Tychonoff

The product of compact spaces is compact (in the product topology).

Properties

• Continuous image of a compact space is compact
• Closed subset of a compact space is compact
• Compact subset of a Hausdorff space is closed