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.