Local homeomorphism from a locally Euclidean space implies Euclidean space

Question

I'm trying to show that for $f:X \rightarrow Y$ local homeomorphism and surjective If $X$ is locally Euclidean so is $Y$.

From Local homeomorphism we have that there exists $V_{x}$ such that $f:V_{x} \rightarrow f(V_{x})$ where $y=f(x)$

But what If the neighbourhood $U_{x}$ from locally Euclidean is not contained in $V_{x}$?

Answer 1

Great! Then take $U_x\cap V_x$.