Compact space
A topological space is a compact space if every open cover has a finite subcover. Thus, if (X) is a topological space and ({U_i}_{i\in I}) is a family of open subsets satisfying
[ X=\bigcup_{i\in I}U_i, ]
then compactness requires the existence of finitely many indices (i_1,\ldots,i_n) for which
[ X=U_{i_1}\cup\cdots\cup U_{i_n}. ]
Compactness is a finiteness condition on the global topology of a space rather than a restriction on the number of its points. A compact space may therefore be infinite, uncountable, or locally indistinguishable from a noncompact space. The definition instead states that no open cover requires infinitely many of its members to cover the entire space.
Some conventions reserve the term compact for spaces that are both compact in the open-cover sense and Hausdorff. Under that convention, a space satisfying only the covering condition is called quasi-compact. This article uses compact without an implicit separation axiom.
Historical development
The concept developed from nineteenth-century results concerning closed and bounded subsets of Euclidean space. The theorem now called the Heine–Borel theorem connected such subsets with finite subcovers by open intervals. Henri Lebesgue subsequently formulated the Lebesgue number lemma, which expressed the uniform scale implicit in an open cover of a compact metric space.
Maurice Fréchet introduced an abstract compactness condition in his 1906 study of metric spaces, initially emphasizing accumulation points and convergent subsequences. In 1924, Pavel Alexandrov, Pavel Urysohn, and You Watanabe incorporated the finite-subcover condition into the emerging language of general topology. Their formulation separated compactness from metric notions such as distance and boundedness, allowing it to be applied to spaces whose topology is not induced by any metric.
The open-cover definition became standard because it remains stable under continuous mappings and arbitrary products. It also applies without countability assumptions, whereas formulations based only on sequences fail to characterize compactness in general topological spaces.
Equivalent formulations
Compactness has several equivalent descriptions that expose different aspects of the same global condition. A family (\mathcal F) of subsets of (X) has the finite intersection property when every finite subfamily has nonempty intersection. The space (X) is compact exactly when every family of closed subsets with this property has nonempty total intersection:
[ \bigcap_{F\in\mathcal F}F\neq\varnothing. ]
This characterization is obtained by taking complements. An open cover without a finite subcover corresponds precisely to a family of closed sets whose finite intersections are nonempty while the intersection of the entire family is empty.
Compactness may also be expressed using filters or nets. A space is compact if and only if every ultrafilter on the space converges to at least one point. Equivalently, every net has a convergent subnet. These formulations replace the countable ordering of a sequence with a general directed structure, thereby detecting topological behavior that sequences alone cannot represent.
In a metric space, the general definitions collapse to several familiar conditions. A metric space is compact if and only if it is sequentially compact, meaning that every sequence has a convergent subsequence. It is also compact if and only if it is both complete and totally bounded. Total boundedness is stronger than ordinary boundedness because it requires a finite cover by balls of every positive radius.
Fundamental properties
The continuous image of a compact space is compact. If (f:X\to Y) is continuous and an open family covers (f(X)), then the inverse images of those open sets cover (X). A finite subcover of (X) consequently produces a finite subcover of (f(X)).
Every closed subspace of a compact space is compact. The converse does not hold without additional assumptions, because a compact subset of a non-Hausdorff space need not be closed. In a Hausdorff space, however, every compact subset is closed. This distinction accounts for the convention in parts of geometry and analysis that incorporates the Hausdorff condition into the definition of compactness.
A continuous bijection from a compact space to a Hausdorff space is a homeomorphism. The map is necessarily closed because compact subsets of the domain have compact images, and compact subsets of the codomain are closed. This result frequently converts a known continuous bijection into a topological equivalence without requiring a separate verification that its inverse is continuous.
Compactness is preserved by arbitrary products. The Tychonoff theorem, associated with Andrey Tychonoff, states that a product
[ \prod_{\alpha\in A}X_\alpha ]
is compact in the product topology whenever every factor (X_\alpha) is compact. For products indexed by an arbitrary set, the theorem has a close logical relationship with the axiom of choice. The corresponding statement generally fails for the finer box topology, whose basic open sets may restrict every coordinate simultaneously.
Compactness in metric and Euclidean spaces
Within (\mathbb R^n) with its usual topology, compactness is equivalent to being closed and bounded. This equivalence is specific to finite-dimensional Euclidean spaces and does not reduce compactness to boundedness in arbitrary topological or metric settings.
The closed interval ([0,1]) is compact, although it contains uncountably many points and admits open covers with infinitely many members. Compactness asserts only that each such cover contains an adequate finite subfamily. By contrast, the interval ((0,1)) is not compact because the missing boundary points permit open covers whose members approach the endpoints without any finite selection covering the entire interval.
An infinite set with the discrete topology is not compact. Its cover by singleton subsets has no finite subcover. A finite discrete space is compact because every cover can be reduced by selecting one covering set for each of its finitely many points.
In an infinite-dimensional normed vector space, a closed and bounded subset need not be compact in the norm topology. In particular, the closed unit ball is compact in that topology exactly when the space is finite-dimensional. The Banach–Alaoglu theorem replaces norm compactness with compactness in the weaker [weak-* topology](/wiki/Weak-* _topology), which is adapted to dual spaces.
Compactification
A compactification of a space (X) is a compact space containing a homeomorphic copy of (X) as a dense subspace. Compactification adds ideal points that encode ways of leaving every compact region of the original space.
Every noncompact, locally compact Hausdorff space has an Alexandrov compactification, also called a one-point compactification. Pavel Alexandrov’s construction adjoins a single point whose neighborhoods are complements of compact closed subsets of the original space. Applying this construction to the real line produces a space homeomorphic to a circle, with the added point representing both unbounded directions through the topology of the compactification.
The Stone–Čech compactification, developed by Marshall Stone and Eduard Čech, is characterized by an extension property for continuous maps into compact Hausdorff spaces. Unlike a one-point compactification, it can add a large collection of points because it records all bounded continuous functions on the original space rather than only its coarse behavior at infinity.
Compactification does not imply that the original space was nearly compact in a metric sense. It expresses the space as a dense part of a compact topological object, while the number and structure of the added points depend on the chosen compactification and its universal properties.
Local compactness
A space is locally compact when each point has a neighborhood related to a compact subspace. For Hausdorff spaces, the standard equivalent formulation requires every point to possess a neighborhood whose closure is compact. Euclidean spaces are locally compact even though they are not compact when their dimension is positive.
Local compactness supports the construction of compactly supported functions and connects general topology with harmonic analysis. It also permits one-point compactification under the Hausdorff assumption. The property remains distinct from compactness because local control around individual points does not supply a finite subcover for the entire space.