Covering Spaces

Definition: Let $X$ be a topological space. A Covering Space of $X$ is a pair $(\tilde{X}, p)$ where: 1. $\tilde{X}$ is a path connected and locally path connected topological space. 2. $p : \tilde{X} \to X$ is a continuous surjective mapping such that for every $x \in X$ there is an path connected open neighbourhood $U$ of $x$ such that the restriction of $p$ on every path component of $p^{-1}(U)$ is a homeomorphism onto $U$. The space $\tilde{X}$ is called a Cover of $X$. The map $p : \tilde{X} \to X$ is called a Covering Map. The open neighbourhoods $U$ of $x$ described above are called Elementary Neighbourhoods.

For example, let $X = S^1$ - the unit circle. Then $\tilde{X} = \mathbb{R}$ is a covering space of $X$. To view this, we wrap $\mathbb{R}$ into a coil and view $X$ as the "shadow" this coil makes.

We will now describe special covering spaces.

Definition: Let $X$ be a topological space. The Trivial Covering Space of $X$ is the covering space consisting of $\tilde{X} = X$ and $p = \mathrm{id}$. A covering space of $X$ is said to be a Universal Covering Space of $X$ if $\pi_1(\tilde{X}, x)$ is the trivial group.

From above we see that the trivial covering space of the circle is the circle, and that $\mathbb{R}$ is a universal covering space of $X$ since $\pi_1(\mathbb{R}, x)$ is the trivial group.