Want to show an embedding is open if the image is open

Question

Let f be an embedding from X to Y. I want to show that if the image of X is open, then f is open. I know that a continious function is open if images of open subsets are open. Futheremore i know that is U $\subset$ X is open, then there exsists some V $\subset$ Y s.t $f^{-1}(V)$=U. So i want to show that f(U)=f($f^{-1}$(V)) is open. An in some way using that i know f(X) is open, i think.

I would be very helpful for any help!

Answer 1

Recall that f:X -> Y is an embedding
when f:X -> f(X) is a homeomorphism.
If U is an open subset of X, then f(U) is open within f(X).
Thus exists open V with f(U) = V $\cap\ $ f(X).
Since f(X) is open, f(U) is open.