A note on finite topologies and switching functions

Abstract Finite topologies and switching functions are investigated. We associate switching functions to families of subsets of a finite set as done for instance by Adam (Truth Functions and the Problem of their Realizations by Two-terminal Graphs [Akademiai Kiado, Budapest, 1968]); we consider then the special case where the families are (finite) topologies. We characterize switching functions which correspond to finite topologies, we associate certain functions, formed with the aid of subfunctions, to topologies on subspaces and on quotient spaces, and we use them to prove some theorems concerning these topologies and to reconstruct (in a certain weak sense) a topological space given the (quotient) spaces obtained by identifying one fixed point with each one of the others.