What is a counter example to this theorem: if $(E,p)$ covering space of connected space $B$, then $|p^{-1}(b)|$ is same for all $b \in B$.

Question

The theorem states that: If $(E,p)$ is a covering space of $B$ being connected space, then all the sets $p^{-1}(b)$, $b\in B$ have the same cardinal number.

I try to find out the counter example if $B$ is not connected but I got stuck. Could you help me point it out?

Answer 1

Take two copies of $D$ (= the unit disk) as your base space : $B = D_1 \sqcup D_2$. Now take $E = D_3 \sqcup D_4 \sqcup D_5$, where your map $p$ send $D_i$ ($i=3,4$) to $D_1$ and $D_5$ to $D_2$ by the identity map. Observe that the fiber of every point is $D_2$ has cardinality 1, and any point in $D_1$ has 2 preimages. That's usually for avoiding such counter-examples we always assume that the base is connected.