Compact space

In mathematics, a compact set is a set for which every covering of that set by a collection of open sets has a finite subcovering. If the set is a subset of a metric space then compactness is equivalent to the set being complete and totally bounded or, equivalently, that every sequence in the set has a convergent subsequence. For the special case that the set is a subset of a finite dimensional normed space, such as the Euclidean spaces, then compactness is equivalent to that set being closed and bounded.

Cover and subcover of a set

Let A be a subset of a set X. A cover for A is any collection of subsets of X whose union contains A. In other words, a cover is of the form

${\mathcal {U}}=\{A_{\gamma }\mid A_{\gamma }\subset X,\,\gamma \in \Gamma \},$

where $\Gamma$ is an arbitrary index set, and satisfies

$A\subset \bigcup _{\gamma \in \Gamma }A_{\gamma }.$

An open cover is a cover in which all of the sets $A_{\gamma }$ are open. Finally, a subcover of ${\mathcal {U}}$ is a subset ${\mathcal {U}}'\subset {\mathcal {U}}$ of the form

${\mathcal {U}}'=\{A_{\gamma }\mid A_{\gamma }\subset X,\,\gamma \in \Gamma '\}$

with $\Gamma '\subset \Gamma$ such that

$A\subset \bigcup _{\gamma \in \Gamma '}A_{\gamma }.$

Formal definition of compact set

A subset A of a set X is said to be compact if every open cover of A has a finite subcover, that is, a subcover which contains at most a finite number of subsets of X (in other words, the index set $\Gamma '$ is finite).