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).