two local homeomorphisms

Question

I am being silly here.

Suppose we have two local homeomorphisms $f: E \to X$ and $g: E' \to X$. If $S$ is a sheet of $E$.

Would $g^{-1}f(E)$ be homeomorphic to $f(E)$? My guess is yes as $g$ is a local homeomorphism. Any help would be appreciated!

Answer 1

Take $X$ to be any nonempty space, then the inclusion $g:\varnothing\hookrightarrow X$ is vacuously a local homeomorphism. Take $f := \operatorname{id}_X:X\to X$, then $f$ is certainly also a local homeomorphism.

Now, $g^{-1}f(X) = \varnothing$ cannot be homeomorphic to $f(X)=X$.