# Universal Covering Space

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.
Well, the natural candidate for that space would be the limit $Y = \varprojlim X_n$, wouldn't it? I haven't checked but it seems to me that the natural maps from $Y$ should be covering maps. – t.b. Sep 12 '11 at 9:23
Dear @Theo, your idea is (as usual) quite interesting and elegant, but unfortunately I don't think it works. Consider the unit circle $S \subset \mathbb C$ and the system of morphisms of coverings of that circle $q_n:S_n=S \to S_{n-1}=S:z\mapsto z^n$ , where the $n$-th covering is given by $p_n:S\to S:z \mapsto z^{n!}$ . The projective limit is a compact (Tychonov!) solenoid, which is not a covering of $S$ and even less its universal cover. – Georges Elencwajg Sep 12 '11 at 11:23