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We apply a categorical lens to the study of betweenness relations by capturing them within a topological category, fibred in lattices, and study several subcategories of it. In particular, we show that its full subcategory of finite objects forms a Fraissé class implying the existence of a countable homogenous betweenness relation. We furthermore show that the subcategory of antisymmetric betweenness relations is reflective. As an application we recover the reflectivity of distributive complete lattices within complete lattices, and we end with some observations on the Dedekind-MacNeille completion.

For a non-empty set X, the collection T op(X) of all topologies on X sits inside the Boolean lattice P(P(X)) (when ordered by set-theoretic inclusion) which in turn can be naturally identified with the Stone space 2 P(X) . Via this identification then, T op(X) naturally inherits the subspace topology from 2 P(X) . Extending ideas of Frink (1942), we apply lattice-theoretic methods to establish an equivalence between the topological closures of sublattices of 2 P(X) and their (completely distributive) completions. We exploit this equivalence when searching for countably infinite compact subsets within T op(X) and in crystalizing the Borel complexity of T op(X). We exhibit infinite compact subsets of T op(X) including, in particular, copies of the Stone-Čech and one-point compactifications of discrete spaces.

Abstract. Premetrics and premetrisable spaces have been long studied and their topological interrelationships are well-understood. Consider the category Pre of premetric spaces and ǫ-δ continuous functions as morphisms. The absence of the triangle inequality implies that the faithful functor Pre → Top -where a premetric space is sent to the topological space it generates -is not full. Moreover, the sequential nature of topological spaces generated from objects in Pre indicates that this functor is not surjective on objects either. Developed from work by Flagg and Weiss, we illustrate an extension Pre ֒→ P together with a faithful and surjective on objects left adjoint functor P → Top as an extension of Pre → Top. We show this represents an optimal scenario given that Pre → Top preserves coproducts only. The objects in P are metric-like objects valued on value distributive lattices whose limits and colimits we show to be generated by free locales on discrete sets.