Kuratowski monoids of $n$-topological spaces

TL;DR

This paper generalizes Kuratowski's 14-set closure-complement theorem to $n$-topological spaces, establishing an upper bound on the number of distinct sets generated by closure and complement operations across multiple topologies.

Contribution

It introduces a generalized framework for Kuratowski monoids in $n$-topological spaces and derives a precise upper bound on the number of distinct sets obtainable.

Findings

Maximum number of distinct sets is $2K(n)$ for $n$-topological spaces.

The bound $2K(n)$ is tight and generalizes the classical Kuratowski result.

The result extends the classical theorem to a broader class of topological structures.

Abstract

Generalizing the famous 14-set closure-complement Theorem of Kuratowski from 1922, we prove that for a set $X$ endowed with $n$ pairwise comparable topologies $\tau_1\subset\dots\subset\tau_n$, by repeated application of the operations of complement and closure in the topologies $\tau_1,\dots,\tau_n$ to a subset $A\subset X$ we can obtains at most $2K(n)=2\sum_{i,j=0}^n\binom{i+j}{i}\binom{i+j}{j}$ distinct sets.