TITLE: Showing the fibre over a point in a covering map is a discrete space.
QUESTION [6 upvotes]: If $ p : \tilde Y \to Y$ is a covering map, how would you show that for every $y \in Y$ we have that $p^{-1}(y)$ is a discrete space?

REPLY [3 votes]: By definition $y$ has an open neighbourhood $U$ such that $p^{-1} [ U ]$ is a disjoint union of open subsets of $\tilde{Y}$, $\{ U_i : i \in I \}$, and $p \restriction U_i : U_i \to U$ is a homeomorphism for all $i$.  These $U_i$ then witness that $p^{-1} [ \{ y \} ]$ is a discrete subspace of $\tilde{Y}$.