Topoloģijas pēc saderīga ideāla moduļa: teorētiskie pētījumi un reprezentācija lokāļu teorijā

Abstract

Abstract The aim of this work is to develop the theoretical foundations of the theory of generalized topological spaces, where the main idea is to perform modulo small sets. The concept of generalized topological space is supported by the corresponding constructions from lattice theory. The main topological notions (interior and closure operators, continuous mapping, weight, density, Lindel of number, product space, etc.) are studied in the framework of generalized topological space. We develop the notion of generalized spatial locale, as an alternative motivation for the concept of generalized topological space, which makes possible to consider the isomorphism between T0 generalized topological spaces and generalized spatial locales and, moreover, to extend the classical duality for all T0 topological spaces without any limitation to sober spaces.