Let \(Y\) have the discrete topology. Show that if \(p: X \times Y \to X\) is projection on the first coordinate, then \(p\) is a covering map.
Solution
Fix \(x\in X\) and let \(U\) be any open neighborhood of \(x\) (for example \(U=X\)). Since \(Y\) is discrete, every singleton \(\{y\}\) is open, and \[ p^{-1}(U)=U\times Y=\bigsqcup_{y\in Y}(U\times\{y\}). \] Each \(U\times\{y\}\) is open in \(X\times Y\), and the restriction \[ p|_{U\times\{y\}}:U\times\{y\}\longrightarrow U \] is a homeomorphism with inverse \(u\mapsto(u,y)\). Thus every open \(U\) is evenly covered; in particular \(p:X\times Y\to X\) is a covering map.