Category of coverings

Locally path-connected, connected covering morphism is a covering

Let and be locally path-connected, connected coverings and be a covering morphism. Then is itself a locally path-connected and connected covering of . homotopy

Proof

Let , and let be a path from to . Further let and be the unique lift of with . Then

and thus both and are lifts of over with , so and in particular . Thus is surjective.

Let . Then has a open neighbourhood that is evenly covered by both and (simply take the intersection of open neighbourhoods with respect to each covering) which we may assume to be connected without loss of generality (otherwise take the connected component containing ). Now let and denote the sheets over in and respectively. By connectedness it follows that for each , for exactly one . Fix some and let as above. It follows

since

hence is a homeomorphism. Clearly , and so from above it follows that the former is some disjoint union of . Therefore is a locally path-connected, connected covering.

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