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Why is $E \sqcup F$ not path connected?

Let $B$ be an arc-connected and locally arc-connected space. Suppose that $p:E\to B$ and $q: F\to B$ are covering spaces. Let $r:E\sqcup F\to B$ be the function such that $r(x)=p(x)$ for all $x\in E$ and $r(x)=q(x)$ for all $x\in F$. Show that $r:E\sqcup F\to B$ is a covering space. “Take $x\in B$, then there is […]