A path connected, locally path connected has a universal covering space implies that the space is semi-locally simply connected

Question

I want to prove A path connected, locally path connected has a universal covering space implies that the space is semi-locally simply connected. Looking forward to some hints.

Answer 1

If $f:E\rightarrow X$ is a covering with $E$ simply-connected, then for every point $x\in X$, it has a neighborhood $U$ which can be mapped homeomorphically to $\widetilde{U}\subset E$, so for any loop $\gamma$ in $\pi_1(U)$, we have a corresponding loop $\widetilde{\gamma}$ in $\widetilde{U}$, which is null-homotopic in $E$, so $\gamma$ is null-homotopic in X.