Metric spaces have distances. Topological spaces don't — they have only open sets. For finite topological spaces, this paper (arXiv: 2603.18580) builds a bridge: a “furtherness” function that plays the role of distance but is derived purely from the topology.
The construction works because finite topological spaces have a unique minimal open set around each point (the intersection of all open sets containing it). Furtherness measures how far apart two points are by how their minimal neighborhoods relate. The resulting function is asymmetric — the furtherness from x to y need not equal that from y to x — because the topology is generally not symmetric (an open set containing x need not contain y, even if one containing y contains x).
The forward balls (neighborhoods in the furtherness metric) recover the original topology. The backward balls recover the opposite topology — the topology obtained by reversing the specialization order. Every finite Tâ space becomes an asymmetric metric space, and the furtherness matrix — the array of all pairwise distances — encodes the complete topological structure.
The through-claim: every finite topology is secretly an asymmetric geometry. The open sets of a finite space contain enough information to define a distance function, but the distance is necessarily directional. Symmetry of distance is a property of Hausdorff spaces; for finite spaces with their coarser topologies, the natural metric is asymmetric. The asymmetry is not a defect — it is the topology speaking in geometric language.
2603.18580. Topology / finite spaces / asymmetric metrics / specialization order / furtherness.