Universal cover

Remarks

Example: $p \colon \R \to S^1$, $p(t) = (\cos 2\pi t, \sin 2\pi t)$, is a covering map. Its total space $\R$ is path-connected and contractible, so $\pi_1(\R) = 1$, and $p$ is a universal cover of the circle. For $n \ge 2$, the quotient map $S^n \to \R P^n$ is a universal cover, because $S^n$ is simply connected.

Non-example: the map $S^1 \to S^1$, $z \mapsto z^2$, on the unit circle in $\C$ is a covering map. Its total space $S^1$ has fundamental group $\Z$, so it is not a universal cover.

If $X$ is connected, locally path-connected and semilocally simply connected, then a universal cover exists. If $X$ is locally path-connected, two universal covers of $X$ are isomorphic over $X$. A universal cover is often written $\tilde X \to X$.