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I'm looking for some intuition regarding universal covers of topological spaces.

$\textbf{Setup:}$ For a topological space $X$ with sufficient adjectives we can construct a/the simply connected covering space of it by looking at equivalence classes of paths at a given base point. We then can put a topology in the standard way done by Hatcher - an open set around an equivalence class of paths, say $[\gamma]$ is the set of $[\gamma\cdot\eta]$ where $\eta$ is a path starting at $\gamma(1)$ contained in $U$ open in $X$.

Here are my questions:

Q: I find this topological space, as constructed above, non-intuitive. Certainly I dont know how I would manipulate it and make topological arguments in it. What is the 'right' way of thinking about the topology here? Or is this construction useful solely for proving the existence of simply connected covers?

Q: Often times it is tractable to construct a simply connected covering by ad-hoc methods (fancy guessing). The projective plane, torus, etc all spring to mind. By universality I know that the covering space obtained by any ad-hoc method is $\textit{the}$ universal covering space obtained by the above method, so there is an isomorphism of these two. Is there a standard way to see this isomorphism? Being really concrete, say in the cases of $\mathbb RP^2$, or $S^1\times S^1$.

In simple terms: how can I 'see' what the universal cover looks like from the general construction?

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There's no reason to expect the universal cover to "look like" anything at all-though when you accomplish an ad hoc construction, it looks an awful lot like the thing you constructed. "Look like" implies there would be something else you've already seen that is homeomorphic or at least homotopic to the abstract universal cover. In general, why should you have seen such a thing? –  Kevin Carlson Oct 18 '12 at 16:13

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I find it nicer to look at all the universal covers at all the base points. One way of doing this it to topologise the fundamental groupoid $\pi_1 X$ (assuming the usual local conditions) by adding neighbourhoods at the two end points of each path class. In this way $\pi_1 X$ becomes a topological groupoid. We have the source and target maps $s,t: \pi_1 X \to X$ and, depending on conventions, $s^{-1}(x)$, or $t^{-1}(x) $, is the universal cover of $X$ based at $x$.

Does this help one to "see" the universal cover?

One possible answer is that to "see" the universal cover one needs an algebraic model of a covering map. This is best supplied, IMHO, by a covering morphism of groupoids. An example of a covering morphism of groupoids is to take a group $G$ and to form the action groupoid for the action of $G$ on itself by multiplication. This gives a groupoid $\widetilde{G}$ with object set the set $G$, and depending on one's conventions, only one arrow $(h,g):hg \to g$ between any two objects, and composition law, $$(k,hg)(h,g)=(kh,g).$$ The covering morphism $\widetilde{G} \to G$ is $(h,g)\mapsto h$. This gives the "universal cover" of the group $G$. For more information, with different conventions, see this book, but that does not do topological groupoids.

Another way of thinking about such spaces is to say that a way of studying a space $X$ is to look at maps $Y \to X$ which are in some ways simpler.

But the origin of covering spaces was in the notion of Riemann surface, and so a way of making complex functions such as $\sqrt{z}$, or $ \log z$, single valued, by defining them on surfaces covering the complex plane minus one or more points. This "back to history" view might really help you to "see" what is wanted and what is going on!

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