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I'm trying a lot to find easy examples of classification theorems of covering spaces of a given space. I've already read some examples here at Mathexchange such as

Classification of covering spaces of $\Bbb{R}\textrm{P}^2 \vee \Bbb{R}\textrm{P}^2$.

Why is a covering space of a torus $T$ homeomorphic either to $\mathbb{R}^2$, $S^1\times\mathbb{R}$ or $T$?

I found these very hard to prove, I would like to know if anyone knows some easy and trivial examples to begin with or if anyone knows some interesting sources to help beginners like me.

Thanks a lot

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I believe Hatcher looks at covers of $S^1 \vee S^1$ in a good bit of detail. Try working through that, then going back to the two-dimensional examples. (If you're not familiar with the book: Allen Hatcher's Algebraic Topology is available for free on his website) – Tabes Bridges Jan 27 '13 at 23:24

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