Compact Space

A compact space is a topological space in which every open cover has a finite subcover. Compactness is a central concept in topology and mathematical analysis: it allows certain problems involving infinitely many local conditions to be reduced to finitely many. Its definition involves only open sets, not distances, coordinates, or a particular shape. In metric spaces, it also has equivalent descriptions involving convergence and completeness. (ocw.mit.edu)

Definition and interpretation

An open cover of a space $X$ is a family $\{U_\alpha\}_{\alpha\in A}$ of open sets whose union equals $X$. The space is compact if, for every such family, there are finitely many indices $\alpha_1,\ldots,\alpha_n$ such that

The original cover may be infinite or uncountable. Compactness requires selecting finitely many members of that particular cover, rather than replacing it with some unrelated finite cover. (ocw.mit.edu)

A subset $K\subseteq X$ is called compact when it is compact with its subspace topology. Equivalently, every family of open subsets of $X$ covering $K$ has a finite subfamily still covering $K$. Thus a compact subset need not itself be open in the surrounding space. (math.ucla.edu)

Euclidean examples and nonexamples

The Heine–Borel theorem states that a subset of Euclidean space $\mathbb R^n$, with its usual topology, is compact exactly when it is closed and bounded. Consequently, closed finite intervals, spheres, and closed bounded balls are compact. The condition of being closed refers to inclusion of boundary and accumulation points; boundedness prevents points from extending arbitrarily far away. (math.mit.edu)

The interval $(0,1)$ is bounded but not compact. For example, the sets

form an open cover of $(0,1)$, but any finite selection leaves some points near zero uncovered. The real line is closed in itself but not compact: the intervals $(-n,n)$, for positive integers $n$, cover it without admitting a finite subcover. These are direct illustrations of why both conditions in Heine–Borel matter. (math.mit.edu)

Every finite topological space is compact, regardless of its topology. In particular, compactness does not imply that a space has infinitely many points or resembles a continuous geometric object. (math.ucla.edu)

Metric characterizations

For a metric space $(X,d)$, the following conditions are equivalent:

• $X$ is compact.
• $X$ is sequentially compact: every sequence has a subsequence converging to a limit in $X$.
• $X$ is a complete metric space and is totally bounded. (maths.ed.ac.uk)

Completeness means that every Cauchy sequence converges in the space. Total boundedness means that, for each $\varepsilon>0$, finitely many open balls of radius $\varepsilon$ cover the space. It is stronger than ordinary boundedness: it demands finite coverage at every arbitrarily small scale. (maths.ed.ac.uk)

For example, an infinite set with the discrete metric, where distinct points have distance $1$, is complete and bounded but not compact. Balls of radius $1/2$ are singletons, so no finite collection covers the space. This shows why “closed and bounded” cannot replace the general metric characterization. (maths.ed.ac.uk)

The metric hypothesis is essential when identifying compactness with sequential compactness. In arbitrary topological spaces, these notions are not equivalent; sequence-based tests alone do not fully characterize compactness. (math.mit.edu)

Preservation and separation properties

A closed subset of a compact space is compact. The image of a compact space under a continuous function is also compact. In particular, compactness is preserved by homeomorphisms, making it an intrinsic topological property rather than a feature of a chosen representation. (math.ucla.edu)

A Hausdorff space is one in which distinct points have disjoint open neighborhoods. Every compact subset of a Hausdorff space is closed. Furthermore, a continuous bijection from a compact space to a Hausdorff space is automatically a homeomorphism. The Hausdorff assumption is important: compact subsets of unrestricted topological spaces need not be closed. (math.ucla.edu)

Products and intersection criteria

Tychonoff’s theorem states that any product of compact spaces is compact in the product topology, including products with infinitely many factors. The specified topology is essential. This theorem extends the elementary finite-product result and is a major tool for constructing compact spaces beyond finite-dimensional geometry. (math.mit.edu)

There is also a closed-set formulation. A space is compact exactly when every family of closed subsets having the finite intersection property has nonempty total intersection. The finite intersection property means that every finite subfamily has nonempty intersection. This characterization follows by taking complements in the open-cover definition. (math.mit.edu)

Consequences for analysis

The extreme value theorem states that a continuous real-valued function on a nonempty compact space is bounded and attains both a maximum and a minimum. Its image is a nonempty compact subset of the real line. In mathematical optimization, this supplies an existence result for a continuous objective function on a nonempty compact feasible set, without requiring differentiability. (math.cmu.edu)

A continuous map from a compact metric space to any metric space is uniformly continuous. Unlike pointwise continuity, uniform continuity provides a single distance threshold that works throughout the domain for each prescribed output tolerance. Compactness therefore strengthens a local regularity condition into a global one. (maths.ed.ac.uk)