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

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 $\mathbb{N}\to\mathbb{N}$.