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 set being complete and totally bounded and again equivalent to sequential compactness: that every sequence in the set 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 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 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 a finite subfamily $\{F_{\lambda _{i}}:i=1,\ldots ,n\}$ 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 to a Hausdorff space 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.