2015/02/28 by Jorge Bruno, J. Bruno, Paul J. Szeptycki +1
Computer Science · Mathematics · #Advanced Algebra and Logic #Advanced Topology and Set Theory #Category of topological spaces #Combinatorics #Computability, Logic, AI Algorithms #Computer science #Coproduct #Discrete mathematics #Distributive property #Extension (predicate logic) #Functional analysis #Functor #Mathematics #Metric space #Morphism #Pure mathematics #Space (punctuation) #Surjective function #Topological space #Topological tensor product #Topology (electrical circuits) #math.CT #math.GN
paper · pdf · doi:10.1007/s10485-016-9465-8
General Topology, Category Theory, Premetrics, premetrizable spaces, free locale, quantales, continuity spaces, premetrics
arxiv created 2016/09/23 · openalex publication_date 2016/09/30 · arxiv updated 2016/11/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Premetrics and premetrisable spaces have been long studied and their topological interrelationships are well-understood. Consider the category \bf Pre of premetric spaces and ε-δ continuous functions as morphisms. The absence of the triangle inequality implies that the faithful functor \bf Pre → \bf 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 \bf Pre indicates that this functor is not surjective on objects either. Developed from work by Flagg and Weiss, we illustrate an extension \bf Pre\hookrightarrow \bf P together with a faithful and surjective on objects left adjoint functor \bf P → \bf Top as an extension of \bf Pre → \bf Top. We show this represents an optimal scenario given that \bf Pre → \bf Top preserves coproducts only. The objects in \bf 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.