Lattice Structures in Finite Graph Topologies

Abstract

In his 2005 dissertation, Antoine Vella studied the relationships between hypergraphs

and a topological space defined using graph-theoretical concepts called the classical

topology. In this paper, we take this process a step further to open set lattices

constructed from these topological spaces. In this paper, we will characterize the

structure of these lattices entirely for finite simple graphs. We then use these results

to conjecture on a possible relationship between hypergraphs and open set lattices.