An automorphism from a connected covering is determined by the image of a point.

Question

Let $p:E \to B$ be a covering map and let $h:E \to E$ be a homeomorphism such that $p=p \circ h$. In this case, it is known that if $E$ is path connected then $h$ is uniquely determined by the image of a point. In Dieck's Algebraic Topology page 65 it is claimed that this same property holds if $E$ is connected, but no proof is provided. Is this really the case?

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There seems no obvious way to prove this directly or by contradiction, and to disprove this I have to find a specific covering of a non-path-connected connected space, which seems pretty hard. So I haven't made any significant progress.

Answer 1

It follows from the following fact:

Let $p: E \to B$ a covering map such that $B$ is connected. Let $f,g: X \to E$ two continuous maps such that $p \circ f = p \circ g$. Then either $f=g$ or $f(x) \neq g(x)$ for every $x \in X$.

This has already been proven in MSE; see for example this answer.

Now if $E$ is connected, in particular so is $B$, so the claim applies.