Continuous lift ($p\tilde{f}=f$)

For continuous maps $p:E\to B$ and $f:X\to B$, a continuous lift of $f$ through $p$ is a continuous map $\tilde{f}:X\to E$ with $p\tilde{f}=f$. Existence and uniqueness require additional hypotheses; the lifting criterion for a covering space states a useful existence criterion when $p$ is a covering map and $X$ is path-connected and locally path-connected.