Heine–Borel theorem

In Euclidean space, a subset is compact if and only if it is closed and bounded. The theorem makes compactness equivalent to two familiar geometric properties.

Compact space: A topological space is compact when every open cover has a finite subcover. The theorem characterizes this general topological property using closedness and boundedness in Euclidean space.
Euclidean space: A real vector space equipped with the standard Euclidean distance and its induced topology. The theorem’s closed-and-bounded criterion applies in finite-dimensional Euclidean spaces.
Closed unit ball in an infinite-dimensional normed space: The closed unit ball of an infinite-dimensional normed space is bounded and closed but not compact. This counterexample shows why closedness and boundedness do not characterize compactness beyond finite dimensions.
Extreme value theorem: A continuous real-valued function on a compact space attains both a maximum and a minimum. The theorem lets closed bounded Euclidean domains meet the compactness hypothesis for extrema.
Open cover: A collection of open sets covers a space when their union contains every point of that space. Finite subcovers are the defining test for compactness used in the theorem.
Metric space: A set equipped with a distance function satisfying nonnegativity, symmetry, and the triangle inequality. Compactness is defined far beyond Euclidean space, where boundedness alone need not suffice.
Compactness in metric spaces: In metric spaces, compactness is equivalent to completeness and total boundedness. This general metric-space criterion replaces the Euclidean closed-and-bounded test.
Uniform continuity theorem: A continuous function on a compact metric space is uniformly continuous. Closed bounded subsets of Euclidean space provide a standard setting for this conclusion.
Closed set: A set is closed when its complement is open, or equivalently in metric spaces when it contains all its limit points. Closedness is one of the two conditions that characterize compact subsets of Euclidean space.
Limit point: A point is a limit point of a set if every neighborhood of it contains another point of the set. A set is closed precisely when it contains all its limit points, including possible subsequential limits.

Linked from 13 pages