How do I see that a homeomorphism is an open function ?

Given a homeomorphism $\sigma: X \rightarrow Y$ between topological spaces, how do i then see that $\sigma(V)$ is open in $Y$ for $V$ open in $X$ ?

I know that $\sigma$ has the following properties: bijection, continuous and continuous inverse $\sigma^{-1}$.

I see that $\sigma^{(-1)}(\sigma(V)) = V$, but does it imply that $\sigma(V)$ is open?


Hint. $\sigma^{-1}$ is continuous, hence preimages of open sets under $\sigma^{-1}$ are open and preimages under $\sigma^{-1}$ are images under $\sigma$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.