The universal cover of a connected topological space X is a simply connected space Y with a map f:Y->X that is a covering map. If X is simply connected, i.e., has a trivial fundamental group, then it is its own universal cover. For instance, the sphere S^2 is its own universal cover. The universal cover is always unique and, under very mild assumptions, always exists. In fact, the universal cover of a topological space X exists iff the space X is connected, locally pathwise-connected, and semilocally simply connected.

covering map | deck transformation | fundamental group | simply connected | universal property