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In a lecture of Algebraic Topology we were given a few examples of universal covering. One of them was the universal covering of connected surfaces with no boundary, saying that its universal covering was the plane (except for the sphere an real proyective plane, which is the sphere).

I don't understand where that comes from, even after asking my professor. Could anyone please help me out?

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    $\begingroup$ What, to you, is a surface? There are two different approaches to answering this question and I'm not sure which you actually want. 1) Cite the classification of surfaces. The only simply connected ones are $\mathbb{R}^2$ and $S^2$ and so only need to argue that $S^2$ only covers $S^2$ and $\mathbb{RP}^2$, hence every other surface is universally covered by $\mathbb{R}^2$. 2) If you have an explicit model of a genus $g$ surface (e.g. as connected sum of $g$ tori, certain quotient of a $4g$-gon), it is possible to explicitly construct a universal cover. $\endgroup$
    – Thorgott
    Commented Mar 11, 2023 at 13:15
  • $\begingroup$ I would like to understand the different types of surfaces a 2-manifolds. If we consider a connected surface with no borders, is it always compact? If so, then we can use the theorem of classification of compact surfaces and check just a few cases. But I'm not sure if there exists non-compact connected surfaces with no boundary. $\endgroup$
    – GreekCorpse
    Commented Mar 11, 2023 at 23:02
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    $\begingroup$ An open disk is certainly non-compact and doesn't have boundary, so no. The claim is true for non-compact surfaces too, but that case is a lot more annoying. Since this is about understanding some examples, I would advise you to only worry about compact surfaces. $\endgroup$
    – Thorgott
    Commented Mar 11, 2023 at 23:40
  • $\begingroup$ Even so, how can I prove that $\Bbb R^2$ is the universal covering of any non-compact connected surface with no border? $\endgroup$
    – GreekCorpse
    Commented Mar 12, 2023 at 16:04
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    $\begingroup$ The universal covering of a non-compact, connected, boundaryless surface is a simply connected, non-compact, connected, boundaryless surface and $\mathbb{R}^2$ is the only such surface. This is a non-trivial result from the classification theory of surfaces. $\endgroup$
    – Thorgott
    Commented Mar 12, 2023 at 16:33

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