`Ultrafilters Relaxed' -- Species of Families of Subsets

Abstract

An assortments of 'species' of families of subsets of a set and some of their properties are investigated, with an eye on the logic and limit role they may play as 'relaxed parallels' to ultrafilters. Introduction, Points vs. FamiliesLet X be a set. For points x ∈ X and subsets S ⊂ X, there is the truth-value whether X ∈ S or not.But with families of subsets F ⊂ P(X), where P(x) is the set of subsets of X, there is also a truth-value -whether S ∈ F or not.Indeed, for every x ∈ X, the family'does the same as x' for that matter: by definition, the truth-value whether x ∈ S is the same as the truth-value whether S ∈ U x .Key words and phrases. families of subsets, ultrafilters and 'relaxing them', logical connectives, filters, eventual families, limits with respect to them, inner and outer (finitely-additive) 'measures', multi-sets and multi-families, common fixed-points, Hausdorff topological spaces.