Lemma: If is a regular covering of X, then for every points of for , there is an equivalence with (link).

If is path-connected and locally path-connected, then there exists a lifting of f which is a closed path by the lifting property (link). Then by the above lemma, every lifting of f should be closed.

I think if is not path-connected, the criteria of "none is closed" is applied along with the above lemma.

Not 100% sure though.