Question

Let f : M → Y be a homeomorphism and O an open subset of M. Explain

concisely in no more than two lines of text why f(O) is an open set

Answer 1

Since $f(O)=(f^{-1})^{-1}(O)$ is a preimage of an open subset $O\subset M$ under a continuous map $f^{-1}:Y\to M,$ we conclude that $f(O)$ is also open.