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

Abstract

In a topological space $(X,\mathcal {T})$ at most 7 distinct sets can be constructed from a set $A \in 2^X$ 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 $(X,\mathcal {T})$ with $|X| \ge 7$ there exists a set with 7 closure-interior relatives; for $|X| < 7$, 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 $(X,\mathcal {T})$ the maximum number of closure-interior relatives of a set is |X|, with one exception: For the indiscrete topology $\mathcal {T}= \{\emptyset ,X\}$ on a set X with $|X|=2$ there exist two sets which possess $|X|+1$ closure-interior relatives.