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
- ∩F = ∅
- 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.
Read more about this topic: Base (topology)
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