A map is a homeomorphism iff it is bijective, continuous, and open

Let and be topological spaces, and be a function. Then is a homeomorphism iff it is bijective, continuous, and open. #m/thm/topology

Proof

A homeomorphism is bijective and continuous by definition. The remaining requirement is that be continuous, which is clearly the case iff maps open sets to open sets, i.e. iff is open.

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