# universal covering space

Let $X$ be a topological space. A universal covering space is a covering space $\tilde{X}$ of $X$ which is connected and simply connected.

If $X$ is based, with basepoint $x$, then a based cover of $X$ is cover of $X$ which is also a based space with a basepoint $x^{\prime}$ such that the covering is a map of based spaces. Note that any cover can be made into a based cover by choosing a basepoint from the pre-images of $x$.

The universal covering space has the following universal property: If $\pi:(\tilde{X},x_{0})\to(X,x)$ is a based universal cover, then for any connected based cover $\pi^{\prime}:(X^{\prime},x^{\prime})\to(X,x)$, there is a unique covering map $\pi^{\prime\prime}:(\tilde{X},x_{0})\to(X^{\prime},x^{\prime})$ such that $\pi=\pi^{\prime}\circ\pi^{\prime\prime}$.

Clearly, if a universal covering exists, it is unique up to unique isomorphism. But not every topological space has a universal cover. In fact $X$ has a universal cover if and only if it is semi-locally simply connected (for example, if it is a locally finite CW-complex or a manifold).

Title universal covering space UniversalCoveringSpace 2013-03-22 12:15:34 2013-03-22 12:15:34 bwebste (988) bwebste (988) 7 bwebste (988) Definition msc 54-00 universal cover OmegaSpectrum