Math and science::Topology

Subbasis

Let $X$ be a set. A subbasis for a topology on $X$ is a collection of subsets of $X$ whose union equals $X$ .

Let $S$ be a subbasis for a topology on $X$ . The set of all unions of finite intersections of elements of $S$ is a topolgy on $X$ . This topology is said to be generated by the subbasis $S$ .

It can be proved that the generated set forms a valid topology.

Basis vs subbasis

A subbasis is an even more lightweight description of a topology compared to a basis. A subbasis can generate a basis for the same topology: the collection of all finite intersections of subbasis elements forms a basis.

Subbasis

Basis vs subbasis

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