Compact topological spaces

Two of the most powerful results that we met when considering functions of a real variable were the Bolzano–Weierstrass theorem and the Heine–Borel theorem. Both of these involve topological properties, and we now consider these properties for topological spaces. We shall see that they give rise to three distinct concepts; in Section 15.4, we shall see that these three are the same for metric spaces.

We begin with compactness; this is the most important of the three properties. It is related to the Heine–Borel theorem, and the definition is essentially the same as for subsets of the real line. If A is a subset of a set X and B is a set of subsets of X then we say that B covers A, or that B is a cover of A, if A ⊆ ⋃B∈BB. A subset C of B is a subcover if it covers A. A cover B is finite if the set B has finitely many members. If (X, τ) is a topological space, then a cover B is open if each B ∈ B is an open set. A topological space (X, τ) is compact if every open cover of X has a finite subcover. A subset A of a topological space (X, τ) is compact if it is compact, with the subspace topology.