The uphill topology of a graph

We define a topology on the vertex set of an undirected graph in which an open set contains, together with each of its vertices, every neighbour of greater or equal degree. Degrees are cardinals, so the definition applies uniformly to finite and infinite graphs. The resulting topology is the Alexandrov topology of finite uphill reachability. We give formulas for minimal open neighbourhoods, closure, interior and boundary; identify the indistinguishable vertices as connected equal-degree plateaus; characterize the T0 and T1 cases, the open and closed points (strict local maxima and minima of the degree), and show that topological components coincide with graph components. We count the open sets for paths, stars and complete bipartite graphs. Examples include graphs with cycles, an empirical social network, a locally finite infinite graph and an infinite star. An extension to loopless multigraphs distinguishes counting incident edges from counting distinct neighbours, and two strict variants illustrate the effect of the treatment of ties.