Problem with definition of covering space

So, Hatcher defines a covering space as

A covering space of a space $X$ is a space $\tilde{X}$ together with a map $p:\tilde{X}→X$ satisfying the following condition: Each point $x\in X$ has an open neighborhood $U$ in $X$ such that $p^{−1} (U)$ is a union of disjoint open sets in $\tilde{X}$ , each of which is mapped homeomorphically onto $U$ by $p$.

Also he says

We allow $p^{−1} (U)$ to be empty, the union of an empty collection of sheets over $U$ , so $p$ need not be surjective.

But then, if the definition is satisfied for every $x \in X$ wouldn't that mean that $p$ maps $\emptyset$ onto some $U$? That doesn't make sense! And if it just means "for all $x \in p(\tilde{X})$" then why say "for all $x \in X$"?

Can someone please explain this to me?

Solution 1:

No, there is no problem and this is why: for every $x\in X$ we need a neighborhood $U$ such that $p^{-1}(U)=\sqcup_{i\in I}\widetilde{U}_i$ for some open sets $\widetilde U_i$ in $\widetilde X$ and some index set $I$. In the case that $p^{-1}(U)$ is empty, we are taking the index set $I$ to be the empty set, rather than having $I$ contain some elements and each $\widetilde U_i$ be the empty set. It's therefore vacuous that each $\widetilde U_i$ map homeomorphically onto $U$, since there are no $i\in I$ to begin with.

As for the second part of your question, you're right that saying "for every $x\in p(\widetilde X)$" would be just as correct. It really just doesn't matter and I guess in the author's head, saying "for every $x\in X$" looks nicer.

Solution 2:

If the base space $X$ is connected, then $p^{-1}(U)=\emptyset$ would indeed imply $p^{-1}(X) =\emptyset$ which is not nonsense, only that $\tilde X$ must be empty.

However, if $X$ is disconnected, any embedding $C\hookrightarrow X$ of a conneceted component is a covering map with empty fires over the other components.