Cardinality Estimations of Sets with Interval Uncertainties in Finite Topological Spaces

TL;DR

This paper develops methods to estimate the size of sets with uncertain boundaries in finite topological spaces, using interval analysis to provide bounds based on topological properties.

Contribution

It introduces new bounds for the cardinalities of sets with interval uncertainties in various classes of finite topological spaces, including non-$T_1$, hyperconnected, and extremely disconnected spaces.

Findings

Interval estimations for set cardinalities based on closure and interior.

New bounds for semi-open sets in specific topological spaces.

Application of interval analysis to topological set cardinalities.

Abstract

In this paper, we have established boundaries of cardinal numbers of nonempty sets in finite non-$T_1$ topological spaces using interval analysis. For a finite set with known cardinality, we give interval estimations based on the closure and interior of the set. In this paper, we give new results for the cardinalities of non-empty semi-open sets in non-$T_1$ topological spaces as well as in extremely disconnected and hyperconnected topological spaces.