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:
- Let $B$ be path connected and semi-locally simply connected. Then $B$ not necessarily has a universal covering.
- Let $B$ be path connected and semi-locally simply connected and have a universal covering. Then $B$ is not necessarily locally path connected.
- 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!