Path lifting theorem

I'm attempting to generalise this theory. Yet, was asking yourself in the evidence offered below and also in a similar way in Hatchers. Can you change the $S^{1}$ with a basic room X. As it appears to not be that vital in the evidence. So if you pick the parts of a basic $X$ can not you make the reduction?

So is $S^{1}$ actually that vital for the evidence? Yet, can not see just how this generalises to a basic cover of $X$, covering it with $\tilde{X}$.