Covering Space Examples - The Torus

Recall from the Covering Spaces page that if $X$ is a topological space then a covering space of $X$ is a pair $(\tilde{X}, p)$ where $\tilde{X}$ is a path connected and locally path connected topological space and $p : \tilde{X} \to X$ is a continuous map such that for every $x \in X$ there exists a path connected open neighbourhood $U$ of $x$ (called an elementary neighbourhood of $x$) such that $p$ restricted to every path component of $p^{-1}(U)$ is a homeomorphism onto $U$.

We will now look at some examples of covering spaces of the torus.

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Let $\tilde{\tilde{X}} = \mathbb{R}$ and let $\tilde{X}$ be the infinite cylinder. Then $\tilde{\tilde{X}}$ is a covering space of $\tilde{X}$:

And if $X = T^2$ is the torus, then $\tilde{X}$ is a covering space of $X$