# Understanding the universal covering of certain surfaces

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).

• 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. Commented Mar 11, 2023 at 13:15
• Even so, how can I prove that $\Bbb R^2$ is the universal covering of any non-compact connected surface with no border? Commented Mar 12, 2023 at 16:04
• 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. Commented Mar 12, 2023 at 16:33