Topological subbasis

Proving open map with a subbasis

Let and each be a topological space, let be a subbasis of , and let . Then is an open map iff the image is open for every subbasic open set . topology

Proof

Clearly if is open the image of every is open. For the converse, first consider the completed basis . Let , implying there exists a finite sequence such that . Then

which is the finite intersection of open sets and is thus open. Hence is open for all . Now consider the whole generated topology . Let , implying there exist such that . Then

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

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