Let $X$ be a homogeneous space (i.e. for every $x,y \in X$ there is a homeomorphism $\phi : X \to X$ such that $\phi(x)=y$) and $x \in X$ a fixed point, if $f$ is a function such that $f(U)$ is open for every $x \in U$ open set, then $f$ is open.
I came across this problem, and I don't know if it is false or true. I don't know how to prove that for every $U$ open set $f(U)$ is open, or maybe I need an extra hypothesis and if someone know it I will thank you very much.