The fiber of a covering space over a connected space has constant cardinality

Let $p: E\to B$ be a covering map ; allow $B$ attached. Show that if $p^{-1}(b_0)$ has $k$ components for some $b_0 \in B$, after that $p^{-1}(b)$ has $k$ components for every single $b \in B$.

I recognize that $E$ has an one-of-a-kind piece due to the fact that $B$ is attached, yet I do not recognize what to do next.

For giving some context, this is Section 53, Exercise 3 of Munkres' Topology.

2022-06-07 14:37:10
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