I've got some doubts about the hypothesis concerning the existence of a universal cover for a space.
More precisely, from Hatcher's book I know that:
Every path connected, locally path connected, semi locally simple connected space admits a universal cover.
Moreover I found out without proof that:
Every path connected, locally contractible space admits an universal cover.
The confusion comes from the fact that we know that if a space is locally contractible then it doesn't necessary implies it to be locally path connected, for example: How to construct a contractible space but not locally path connected?
Is there something wrong with the second statement? If not, could somebody point to a proof of it?
Thanks
Suppose that $X$ is path-connected and locally contractible.
Then let $x$ be in $X$, and $O$ any open set containing $x$, and let $U_x$ be a contractible neighbourhood of $x$ such that $U_x \subseteq O$.
As a contractible space is path-connected, $U_x$ is path-connected. It follows that $X$ is locally path-connected. Also, any loop on inside $U_x$ starting in $x$ can be continuously be contracted to $x$, by a homotopy $U_x \times [0,1] \to U_x$, as $U_x$ is contractible. So by definition (e.g. see Wikipedia, we only require the homotopy to be defined with codomain $X$) $X$ is semi-locally simply connected.
Now the quoted theorem in Hatcher allows us to state that $X$ has a universal cover.