Is E necessarily connected if $p: E \to B$ is a finite covering map and $B$ is connected and semi locally simply connected?

Question

Is E necessarily connected if $p: E \to B$ is a finite covering map (i.e $p^{-1}(b_0)$ is finite) and $B$ is connected and semi locally simply connected? What if $B$ is also locally path connected?

From this question, I know that if $B$ is locally (path-)connected and $p:E\to B$ is a covering map, then $E$ is locally (path-) connected.

I am trying to prove that $E$ is connected.

Answer 1

What about the projection $\pi_1: \Bbb R \times \{0,1\} \to \Bbb R$? This is a finite covering map, and the codomain is locally path-connected and path-connected (and semi-locally simply connected) and the domain is not (path-)connected.