A covering is said to be regular if for some the group is a normal subgroup of . Prove that if is a closed path in then either every lifting of is closed or none is closed.
A covering is said to be regular if for some the group is a normal subgroup of . Prove that if is a closed path in then either every lifting of is closed or none is closed.
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.