On supratopologies, normalized families and Frankl conjecture

TL;DR

This paper introduces generalized topological concepts to analyze union-closed families, reducing Frankl's conjecture to supratopological spaces and proving it for certain classes via a novel reduction method.

Contribution

It develops a new framework using supratopological spaces and normalized families to approach Frankl's conjecture, including a reduction technique and a proof for a specific family class.

Findings

Reduction of Frankl's conjecture to supratopological spaces.

A new method to reduce normalized families via dual families.

Proof of Frankl's conjecture for families obtained through iterative reduction.

Abstract

We introduce some generalized topological concepts to deal with union-closed families, and show that one can reduce the proof of Frankl's conjecture to some families of so-called supratopological spaces. We prove some results on the structure of normalized families, presenting a new way of reducing such a family to a smaller one using dual families. Applying our reduction method, we prove a refinement of a conjecture originally proposed by Poonen. Finally, we show that Frankl's Conjecture holds for the class of families obtained from successively applying the reduction process to a power set.