Let \(X\) be a set. A topology on \(X\) is a collection \(\tau\) of subsets of \(X\) such that
\(\emptyset \in \tau\) and \(X\in \tau\);
\(\Union_{i\in I} U_i \in \tau\) for every family \((U_i)_{i\in I}\) of elements of \(\tau\);
\(U_1\intersect \cdots \intersect U_n \in \tau\) for all \(n\geq 1\) and \(U_1, \ldots, U_n\in\tau\).
The pair \((X, \tau)\) is a topological space, the elements of \(\tau\) are its open sets, and a subset \(F\subseteq X\) is closed if \(X\sm F\) is open.
By De Morgan’s laws, a collection \(\mathcal F\) of subsets of \(X\) is the collection of closed sets of a topology on \(X\) if and only if \(\emptyset, X\in\mathcal F\), \(\mathcal F\) is closed under arbitrary intersections, and \(\mathcal F\) is closed under finite unions; the topology is then \(\ts{X\sm F \st F\in\mathcal F}\).