Let $X,Y,Z$ be connected topological spaces, $f\colon X\to Y$ be a continuous map and $p\colon Z\to Y$ be a covering map. The problem is the existence of a continuous lift of $f$ across $p$. A standard result inovlving fundamental groups and induced homomorphisms requires that $X$ be path-connected and locally path-connected. Sufficient conditions however exist also in case of not necessarily locally path-connected spaces $X$. Say, if $X$ is contractible then the lift does exist. Could you please recommend a work containing sufficient conditions for more general spaces $X$ than the locally path-connected ones? (Possibly with further restrictions on $f$ or $p$.)