Compact spaces and spaces of maximal complete subgraphs

Abstract

We consider the space M ( G ) M(G) of all maximal complete subgraphs of a graph G G and, in particular, the space M ( P ) M(P) of all maximal chains of an ordered set P P . The main question considered is the following: Which compact spaces can be represented as M ( G ) M(G) for some graph G G or as M ( P ) M(P) for some ordered set P P ? The former are characterized as spaces which have a binary subbase for the closed sets which consists of clopen sets. We give an example to show that this does not include all zero-dimensional supercompact spaces. The following negative result is obtained concerning ordered sets: Let D D be an uncountable discrete space and let α D \alpha D denote the one-point compactification of D D . Then there is no ordered set P P such that M ( P ) ≃ α D M(P) \simeq \alpha D .