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.

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- Apr 25th 2010, 02:55 PMJJMC89[SOLVED] Algebraic Topology: Regular Cover
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.

- Apr 25th 2010, 11:24 PMaliceinwonderland
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.