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

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