there exists some such that .Property (B1) corresponds to the fact that is an open set; property (B2) corresponds to the fact that is an open set.

Conversely, suppose is just a set without any topology and is a family of subsets of satisfying properties (B1) and (B2). Then is a base for the topology that it generates. More precisely, let be the family of all subsets of that are unions of subfamilies of Then is a topology on and is a base for .(Sketch: defines a topology because it is stable under arbitrary unions by construction,