Let $X, Y$ be topological spaces and $f : X \to Y$ be continuous. At any point $p \in Y$, we have the fiber $F = f^{-1}(p)$ over $p$ and the stalk $S$ whose elements are germs at $p$ of sections of $f$. There is an evident mapping $f' : S \to F$ which sends each germ to its value at $p$.
If $f$ is a local homeomorphism, then each $f'$ is a bijection. Is the converse true?
No. For instance, let $X$ be any space and $Y$ be a single point. Then every $f'$ is a bijection, but $f$ is not a local homeomorphism unless $X$ is discrete.
A bit less trivially, let $Y$ be a locally connected space, let $D$ be a totally disconnected space, let $X=Y\times D$, and let $f$ be the projection. Then each $f'$ is a bijection, since any section of $f$ must be locally constant on the $D$ coordinate by connectedness. But $f$ is not a local homeomorphism unless $D$ is discrete.