Topological subbasis

Proving continuity with a subbasis

Let and each be a Topological space, let be a subbasis of , and let . Then is continuous iff the preïmage is open for all . #m/thm/topology

Proof

Since a subbasis is a family of open sets, it is clear that given continuous the preïmage of subbasic open neighbourhoods is open. Let such that for all , the preïmage is open. First consider the completed basis . Let , implying there exists a finite sequence where such that . Then

which is the finite intersection of open sets and is thus open. Hence for all , the preïmage is open. Now consider the entire generated topology . Let , implying there exists an indexed family such that . Then

which is the union of open sets and thus open. Hence the preïmage of every open set is open, wherefore is continuous.

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