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

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 $$\mathbb{N}$\to $\mathbb{N}$$withfinitefibers,oronthesubmonoid$$\mathbb{E}$\subseteq $\mathbb{M}$$ofthemonotoneandinjectivemaps$$\mathbb{N}$\to$\mathbb{N}$$.