Classification of covering spaces for spaces that are not locally path connected: counterexamples?

Question

The standard theory treats the case where the base space $B$ is path connected, locally path connected, and semi-locally simply connected. While being path connected and semi-locally simply connected is necessary to have a universal covering (which by definitions is just a covering with simply connected total space), the condition local path connectedness is not so natural. So I'd like to see counterexamples for the following:

1. Let $B$ be path connected and semi-locally simply connected. Then $B$ not necessarily has a universal covering.
2. Let $B$ be path connected and semi-locally simply connected and have a universal covering. Then $B$ is not necessarily locally path connected.
3. Let $B$ be path connected and semi-locally simply connected and have a universal covering $p:E\to B$. Do we still have the usual theory that connected coverings of $B$ correspond to subgroups of $\pi_1(B)$? In particular, is the group of deck transformations of $E$ isomorphic to the group $\pi_1(B)$?

Thanks in advance!

Answer 1

Here is a partial answer.

2. Let $X$ be any path connected simply connected space which is not locally connected (which implies that it is not locally path connected). As an example take the Warsaw circle (see https://de.wikipedia.org/wiki/Datei:Warsaw_Circle.png, https://en.wikipedia.org/wiki/Shape_theory_(mathematics), How to show Warsaw circle is non-contractible?). Then $id : X \to X$ is a universal covering.

Answer 2

Here is another partial answer.

3. As in the answer to 2., let $W$ be the Warsaw circle which is path connected simply connected. It has a universal covering, $id : W \to W$. However, it has infinitely many distinct connected coverings, and these cannot be classified by subgroups of $\pi_1(W) = 0$. These coverings are obtained by pasting together $n$ copies of the closed toplogist's sine curve $S$ into a "circular" pattern and mapping this space in the obvious way to $W$ by wrapping it $n$-times around $W$. Another covering is obtained by pasting together infinitely copies of $S$ into a "linear" pattern and mapping this space in the obvious way to $W$. This is in complete analogy to the coverings $z^n : S^1 \to S^1$ and $e^{2\pi it} : \mathbb{R} \to S^1$.

Note that all these coverings are not path connected. It therefore potentially makes a difference whether we work with connected coverings or with path connected coverings. For a locally path connected base space $X$ this is the same.