Base (topology) - Theorems

Theorems

• For each point x in an open set U, there is a base element containing x and contained in U.
• A topology T₂ is finer than a topology T₁ if and only if for each x and each base element B of T₁ containing x, there is a base element of T₂ containing x and contained in B.
• If B₁,B₂,...,B_n are bases for the topologies T₁,T₂,...,T_n, then the set product B₁ × B₂ × ... × B_n is a base for the product topology T₁ × T₂ × ... × T_n. In the case of an infinite product, this still applies, except that all but finitely many of the base elements must be the entire space.
• Let B be a base for X and let Y be a subspace of X. Then if we intersect each element of B with Y, the resulting collection of sets is a base for the subspace Y.
• If a function f:X → Y maps every base element of X into an open set of Y, it is an open map. Similarly, if every preimage of a base element of Y is open in X, then f is continuous.
• A collection of subsets of X is a topology on X if and only if it generates itself.
• B is a basis for a topological space X if and only if the subcollection of elements of B which contain x form a local base at x, for any point x of X.