Topology MOC

Topological subbasis

Any family of subsets S ⊆P(𝑋) whose union ⋃𝑆∈S𝑆 =𝑋 forms a subbasis for a topology. The generated topology is the coarsest topology containing S. topology A stronger concept is the Topological basis, which can be formed by adding all finite intersections of subbasic open neighbourhoods. topology

Proof the generated topology is well defined and matches the basis

Let S ⊆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

• Proving continuity with a subbasis
• Proving open map with a subbasis
• Alexander subbasis theorem

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