Graphs on Topological Spaces

Abstract

Graphs defined on groups, such as power graphs, enhanced power graphs and commuting graphs, form a hierarchy under inclusion of edge sets. We develop an analogous theory for topological spaces. To a space X we attach five graphs on the vertex set X, all invariant under homeomorphisms: the specialization, inseparability, closure-meeting, Urysohn and functional graphs. They form a diamond-shaped hierarchy whose members are null exactly when X is T_1, Hausdorff, Urysohn or functionally Hausdorff. The Urysohn graph always contains the square of the inseparability graph, with equality for Alexandrov spaces, and the hierarchy is compatible with products via the strong product of graphs. Every countable graph, together with isolated vertices, is the inseparability graph of a second countable T_1 space. For finite spaces we characterize the graphs that occur as inseparability graphs (every edge lies in the closed neighbourhood of a simplicial vertex), determine all coincidences between members of the hierarchy, and show that the specialization graph determines the weak homotopy type but not the space itself. Using Dowker duality we prove that the complexes of inseparable and of closure-meeting families are always homotopy equivalent and, under an acyclicity hypothesis, recover the weak homotopy type. We show that if the closure-meeting graph is a cograph then it is chordal, an analogue of a recent result for enhanced power graphs; the same holds for the inseparability graph of an Alexandrov space but not in general. Finally, following Cameron's work on topologies preserved by permutation groups, we show that for a topology invariant under a primitive group each graph is null or connected, and that the neighbourhood filters of the five graphs reverse the hierarchy, giving spaces whose specialization graph contains the Rado graph as a spanning subgraph.