= Graph structure lifted through a covering map
A <covering map> over a <topological graph> lifts its <vertices> to all their preimages and each open <edge> to disjoint open <edges>. Their endpoints are supplied by the <path lifting theorem> applied to each parametrizing <path> of a closed <edge>. The <covering sheets> over small vertex stars verify that the resulting <graph> topology agrees with the original <topology> upstairs. A constant finite <degree of a covering map> $d$ therefore multiplies both vertex and edge counts by $d$, proving the <Euler characteristic under a finite covering> formula for finite <topological graphs>.
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