Topological subbasis

Any family of subsets $S \subseteq P (𝑋)$ whose union $⋃ 𝑆 \in S 𝑆 = 𝑋$ forms a subbasis for a topology. The generated topology is the coarsest topology containing $S$ . #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 $S \subseteq P (𝑋)$ be a family of subsets whose union equals $𝑋$ We claim that there exists a coarsest topology $T$ containing $S$ . In order to satisfy the axioms for a Topological space, $T$ must be closed under finite intersection and (in)finite union. If we first complete $S$ under finite intersection to obtain a Topological basis $B$ , and thereafter under (in)finite union, we obtain a complete $T$ , since the finite intersection of unions may always be expressed as the union of finite intersections.

Properties

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