Let be a topological space. A universal covering space is a covering space of which is connected and simply connected.

If is based, with basepoint , then a based cover of is cover of which is also a based space with a basepoint 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 .

The universal covering space has the following universal property: If is a based universal cover, then for any connected based cover , there is a unique covering map such that .

Clearly, if a universal covering exists, it is unique up to unique isomorphism. But not every topological space has a universal cover. In fact 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).