If $X$ is path connected, locally path connected and semi-locally simply connected topological space and $x_0\in X$, consider the set $\chi =\{(X_{\alpha},x_\alpha) \} $ of covering spaces of $X$ with covering map $p_\alpha \colon X_\alpha \rightarrow X$, $p_\alpha (x_\alpha)=x_0$. (Clearly, this is a non-empty collection, $(X,x_0)\in \chi$ )
Put a partial ordering relation on this collection: $(X_\alpha,x_\alpha)\geq (X_\beta,x_\beta)$, if there is a covering map $q\colon X_\alpha \rightarrow X_\beta$ such that $p_\beta \circ q =p_\alpha$.
Can we use Zorn's lemma to prove existence of universal covering space of $X$ as maximal element of this collection?
If yes, are the conditions stated (path connected, etc.) in hypothesis are necessary (they are necessary when we prove existence, in usual topological way)?