Definition: Let X be a topological space. A subset F of X is said to be a closed set if and only if its complement X F is an open set.

We recall that the complement of the union of some collection of subsets of some set X is the intersection of the complements of those sets, and the complement of the intersection of some collection of subsets of X is the union of the complements of those sets. The following result therefore follows directly from the definition of a topological space.

Proposition 1.1 Let X be a topological space. Then the collection of closed sets of X has the following properties:—

(i) the empty set ∅ and the whole set X are closed sets,

(ii) the intersection of any collection of closed sets is itself a closed set,