I am reading Chapter 13, the chapter about classification of covering spaces, of J.Munkres' Topology. My confusion raised when I read Corollary 82.2. which says:
the space $B$ has a universal covering space if and only if $B$ is path connected, locally path connected, and semilocally simply connected.
The book does not give a proof so I believe it should be straight forward. But I just can not prove the "only if" part of this corollary. I do not know how to see that a space which has a universal cover must be locally path connected. I understand that by Lemma 80.4., a base space with a universal cover has to be semilocally simply connected. And since covering space is simply connected, the base space must be path connected.
Thank you very much for your attention and really appreciate your helps.