Compact space

In mathematics, a compact space is a topological space for which every covering of that space by a collection of open sets has a finite subcovering. If the space is a metric space then compactness is equivalent to the space being complete and totally bounded and again equivalent to sequential compactness: that every sequence in this space has a convergent subsequence.

A subset of a topological space is compact if it is compact with respect to the subspace topology. A compact subset of a Hausdorff space is closed, but the converse does not hold in general. 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: this is the Heine-Borel theorem.

Cover and subcover of a set
Let A be a subset of a set X. A cover for A is any family of subsets of X whose union contains A. In other words, a cover is of the form
 * $$\mathcal{U}=\{A_{\gamma} \mid \gamma \in \Gamma \}, \quad A_{\gamma} \subset X, $$

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 family $$\mathcal{U}'$$ of the form
 * $$\mathcal{U}'=\{A_{\gamma} \mid \gamma \in \Gamma'\}$$

with $$\Gamma' \subset \Gamma$$ such that
 * $$A \subset \bigcup_{\gamma \in \Gamma'}A_{\gamma}.$$

Formal definition of compact space
A topological space X is said to be compact if every open cover of X 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).

Finite intersection property
Just as the topology on a topological space may be defined in terms of the closed sets rather than the open sets, so we may transpose the definition of compactness in terms of open sets into a definition in terms of closed sets. A space is compact if the closed sets have the finite intersection property: if $$\{ F_\lambda : \lambda \in \Lambda \}$$ is a family of closed sets with empty intersection, $$\bigcap_{\lambda \in \Lambda} F_\lambda = \emptyset$$, then there exists a finite subfamily $$\{ F_{\lambda_i} : i=1,\ldots,n \}$$ that has empty intersection, $$\bigcap_{i=1}^n F_{\lambda_i} = \emptyset$$.

Examples

 * Any finite space.
 * An indiscrete space.
 * A space with the cofinite topology.
 * The Heine-Borel theorem: In Euclidean space with the usual topology, a subset is compact if and only if it is closed and bounded.

Properties

 * Compactness is a topological invariant: that is, a topological space homeomorphic to a compact space is again compact.
 * A closed set in a compact space is again compact.
 * A subset of a Hausdorff space which is compact (with the subspace topology) is closed.
 * The quotient topology on an image of a compact space is compact
 * The image of a compact space under a continuous map is compact.
 * A continuous real-valued function on a compact space is bounded and attains its bounds.
 * The Cartesian product of two (and hence finitely many) compact spaces with the product topology is compact.
 * The Tychonoff product theorem: The product of any family of compact spaces with the product topology is compact. This is equivalent to the Axiom of Choice.
 * If a space is both compact and Hausdorff then no finer topology on the space is compact, and no coarser topology is Hausdorff.