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

Let $𝑋$ and $𝑌$ be topological spaces, and $𝑓 : 𝑋 \to 𝑌$ be a function. Then $𝑓$ 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 $𝑓 - 1$ be continuous, which is clearly the case iff $𝑓$ maps open sets to open sets, i.e. iff $𝑓$ is open.

#state/tidy | #lang/en | #SemBr