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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)?

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One cannot prove this using Zorn's lemma, at least not in an obvious way. The problem is that I don't see how to show that if $(X_1,x_1) < (X_2,x_2) < \ldots$ is a totally ordered sequence, then there exists some upper bound $(Y,y)$ with $(X_i,x_i) \leq (Y,y)$ for all $i$. You have to come up with a space somehow, and I don't think you'll be able to bypass the usual construction.

It should be noted that the conditions "locally path connected" and "semilocally simply connected" are actually necessary for your space to have a universal cover (exercise : prove this!).

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    @t.b. , your idea is not the right way to construct the universal cover, but it is useful and interesting! If $X$ is a graph, say, then your limit $\varprojlim X_n$ is a profinite tree, and the profinite fundamental group of $X$, which is a free profinite group, acts on it. Ribes and Zalesskii have developed a very nice theory of profinite trees.2013-10-21