Compact Set

Overview

Many early theorems of topology made statements about closed and bounded subsets of the real numbers. It was later discovered that the key to these theorems was the more abstract notion of compactness.

Definition

A topological space {% M %} is called compact if every open cover of {% M %} contains a finite set of open sets that cover {% M %}

Here an open cover of a topological space {% M %} is defined to be a set of open subsets {% \{ O_i \subset M \} %} such that {% \cup_i O_i = M %}