Cofinite subsets and double negation topologies on locales of filters and ideals

TL;DR

This paper investigates the structure of cofinite subsets within the locale of filters on a set, using double negation topologies, and characterizes cofinite subsets in the case of natural numbers through monoid structures.

Contribution

It introduces a novel analysis of cofinite subsets via double negation topologies on filter locales and characterizes these subsets for natural numbers using monoid morphisms.

Findings

Cofinite subsets are characterized by double negation topology on filter locales.

The paper establishes an essential locale morphism from the power set to the filter locale.

Cofinite subsets of natural numbers are characterized via monoid structures of functions with finite fibers.

Abstract

We study the role of the filter $c\mathcal{K}(X)$ of cofinite subsets of $X$ in the locale $\mathcal{F}ilt(X)$ of all filters on $X$, by means of the double negation topology of $\mathcal{F}ilt(X)$, and an essential locale morphism $\mathcal{P}(X)^{op}\to\mathcal{F}ilt(X)$. Moreover, in the case $X=\mathbb{N}$, we characterise cofinite subsets by means of the double negation topology on the monoid $\mathbb{M}$ of the maps $\mathbb{N}\to \mathbb{N}$ with finite fibers, or on the submonoid $\mathbb{E}\subseteq \mathbb{M}$ of the monotone and injective maps $\mathbb{N}\to\mathbb{N}$.