Tool-Based Relational Investigation of Closure-Interior Relatives for Finite Topological Spaces

In a topological space (X, T) at most 7 distinct sets can be constructed from a set A is an element of 2(X) by successive applications of the closure and interior operation in any order. If sets so constructed are called closure-interior relatives of A, then for each topological space (X, T) with vertical bar X vertical bar >= 7 there exists a set with 7 closure-interior relatives; for vertical bar X vertical bar < 7, however, 7 closure-interior relatives of a set cannot co-exist. Using relation algebra and the RelView tool we compute all closure-interior relatives for all topological spaces with less than 7 points. From these results we obtain that for all finite topological spaces (X, T) the maximum number of closure-interior relatives of a set is vertical bar X vertical bar, with one exception: For the indiscrete topology T = {empyt set , X} on a set X with vertical bar X vertical bar = 2 there exist two sets which possess vertical bar X vertical bar + 1 closure-interior relatives.