Base (topology) - Base For The Closed Sets

Base For The Closed Sets

Closed sets are equally adept at describing the topology of a space. There is, therefore, a dual notion of a base for the closed sets of a topological space. Given a topological space X, a base for the closed sets of X is a family of closed sets F such that any closed set A is an intersection of members of F.

Equivalently, a family of closed sets forms a base for the closed sets if for each closed set A and each point x not in A there exists an element of F containing A but not containing x.

It is easy to check that F is a base for the closed sets of X if and only if the family of complements of members of F is a base for the open sets of X.

Let F be a base for the closed sets of X. Then

1. ∩F = ∅
2. For each F₁ and F₂ in F the union F₁ ∪ F₂ is the intersection of some subfamily of F (i.e. for any x not in F₁ or F₂ there is an F₃ in F containing F₁ ∪ F₂ and not containing x).

Any collection of subsets of a set X satisfying these properties forms a base for the closed sets of a topology on X. The closed sets of this topology are precisely the intersections of members of F.

In some cases it is more convenient to use a base for the closed sets rather than the open ones. For example, a space is completely regular if and only if the zero sets form a base for the closed sets. Given any topological space X, the zero sets form the base for the closed sets of some topology on X. This topology will be finest completely regular topology on X coarser than the original one. In a similar vein, the Zariski topology on An is defined by taking the zero sets of polynomial functions as a base for the closed sets.