Finite Topological Spaces: Coarser and Finer Structures with Connections to Graph Theory

Abstract

Finite topological spaces provide a tractable setting for studying fundamental topological properties and their interactions with other mathematical structures. In this paper, we focus exclusively on finite topological spaces, emphasizing the role of coarser and finer topologies, as well as the extremal cases of discrete and indiscrete topologies. Special attention is given to T₀​ (Kolmogorov) spaces, which are central in the finite context, since any finite T₁​ space is necessarily discrete. We also highlight the significance of the Sierpiński space as the smallest non-discrete and non-indiscrete topology, noting its importance as a classifying space for open sets and its connections to semantics and computational theory. The study further situates finite topological spaces within ongoing research that links topology and graph theory. Various constructions of topologies derived from graph-theoretic concepts are reviewed, including those based on closed neighborhoods, subbases, open hop neighborhoods, and monophonic eccentric neighborhoods. These approaches have led to characterizations of graphs that induce discrete or indiscrete topologies, as well as to the development of compatible topologies where graph connectivity corresponds to topological connectedness. Recent investigations into discrete topological graphs, domination in discrete topological spaces, and special intersection graphs are also discussed. Collectively, these works demonstrate the growing interplay between finite topology and graph theory and motivate further exploration of their combined structures and applications.