Compact space

Equivalence of definitions
The equivalence with definition (4) follows from the Alexander subbase theorem.

In the real line and Euclidean space

 * Any interval of the form $$[a,b]$$ (with both $$a$$ and $$b$$ real numbers) is a compact space, with the subspace topology inherited from the usual topology on the real line. More generally, any finite union of such intervals is compact.
 * Compact subsets could look very different from unions of intervals. For instance, the Cantor set is compact.
 * A subset of the real line, or more generally, of Euclidean space, is compact with the subspace topology if and only if it is closed and bounded (i.e., it can be enclosed inside some large enough ball). See Heine-Borel theorem
 * Note that it is not true for arbitrary metric spaces that closed and bounded subsets are compact. In fact, for normed real and complex vector spaces, that occur extensively in functional analysis, closed and bounded iff compact is equivalent to being finite-dimensional. Much of the difficulty and challenge of dealing with infinite-dimensional normed real and complex vector spaces is coming up with conditions analogous to compactness that allow reasoning similar to that done in the finite-dimensional case.

More general examples

 * For a metric space to be compact with the induced topology is equivalent to a condition on it called being totally bounded. See compact metric space.
 * The geometric realization of any finite simplicial complex is a compact space. (Geometric realizations of simplicial complexes are called polyhedra). See compact polyhedron.
 * The geometric realiation of a CW-complex with finitely many cells is a compact space. (Geometric realizations of CW-complexes are termed CW-spaces).
 * Some (but not all) manifolds are compact manifolds, and much of the theory of manifolds relies on the crucial distinction between compact and non-compact manifolds.

In commutative algebra
The spectrum of a commutative unital ring, equipped with the Zariski topology, is always compact (though almost never Hausdorff).

Conjunction with other properties

 * Compact Hausdorff space: Conjunction with the property of being a Hausdorff space
 * Compact manifold: Conjunction with the property of being a manifold
 * Compact metrizable space: Conjunction with the property of being a metrizable space

Refinement formal expression
In the refinement formalism, the property of compactness has the following refinement formal expression:

Open $$\to$$ Finite open

Textbook references

 * , Page 164 (formal definition)
 * , Page 12 (formal definition)