$f$ has to be open in order for continuous bijection to be a homeomorphism?

Question

Claim :

Let $f: X\to Y$ be a continuous bijection, then $f$ is a homeomorphism if $f$ is open

Is the claim true? I thought $f$ was a homemorphism if $f^{-1}$ is a continuous function

Answer 1

If $f$ is a continuous bijection and is open, then for an open set $U$, $(f^{-1})^{-1}(U)=f(U)$ is open. Thus $f^{-1}$ is continuous so $f$ is a homeomorphism. (Usually I wouldn't give the answer but the whole problem is just one step...)