[[Homeomorphism]]
# 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 

> [!check]- 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.
> <span class="QED"/>

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