Topology MOC

Topological subbasis

Any family of subsets whose union forms a subbasis for a topology. The generated topology is the coarsest topology containing . #m/def/topology A stronger concept is the Topological basis, which can be formed by adding all finite intersections of subbasic open neighbourhoods. #m/thm/topology

Proof the generated topology is well defined and matches the basis

Let be a family of subsets whose union equals We claim that there exists a coarsest topology containing . In order to satisfy the axioms for a Topological space, must be closed under finite intersection and (in)finite union. If we first complete under finite intersection to obtain a Topological basis , and thereafter under (in)finite union, we obtain a complete , since the finite intersection of unions may always be expressed as the union of finite intersections.

Properties

#state/tidy | #lang/en | #SemBr