Topology, A Local Base for the Topology

A Local Base for the Topology

One can define base locally, at a given point. A collection of base open sets is a base at p if every open set containing p contains one of these base sets, which contains p. You don't need the entire base for the topology, and you may not even need all the base sets that contain p.

Let's show that B is a base for a given topology iff it includes a base at p for every point p. Given the former, place p in an arbitrary open set W, which is the union of base sets, hence p is contained in a base open set inside W. Conversely, if every p has a base, cover W with base open sets containing the points of W, and contained in W, and W becomes the union of base open sets, as is required by a base. If p is in the intersection of base open sets, p is contained in a base open set that is contained in the intersection, and that takes care of the base criterion.