Filters, nets and cofinal types

Abstract

In this paper we investigate functionals relating filters and nets on a given set X, with special respect to the problem of monotonicity. In particular, we provide three different functionals $\Psi_{k}\left(k=1,2,3\right)$ from the collection of the filters on X to the class of the nets of X, such that if $\mathcal{F}\supseteq\mathcal{G}$ then $\Psi_{k}\left(\mathcal{F}\right)$ is a subnet of $\Psi_{k}\left(\mathcal{G}\right)$. We also compare them with the standard functional N, which fails to be monotone. To this end, we often use the theory of cofinal types.