General Topology as Ordered Semigroup Actions: Book 1

Algebraic General Topology, Volume 1, unpacks the foundations of algebraic general topology, introducing novel concepts and exploring their applications.

The focus lies on two key advancements: funcoids and reloids. Funcoids offer a broader perspective on proximity spaces, while reloids provide a refined understanding of generalized uniform spaces. Both concepts extend traditional binary relations by employing filters instead of sets within their domains and ranges. This allows them to act as a unifying framework, encompassing the fields of calculus and discrete mathematics through a shared foundation.

In Algebraic General Topology, author Victor Lvovich Porton establishes a novel definition of continuity, expressed through an elegant algebraic formula. This definition applies to various morphisms (including funcoids and reloids) within the context of partially ordered categories. Notably, it encompasses traditional continuity, proximity continuity, and uniform continuity within a single, unifying formula. Additionally, the concept of connectedness is introduced specifically for funcoids and reloids.

Further exploration discusses generalizations of funcoids, including point-free funcoids, staroids, and multifuncoids. The book also examines various product constructions applicable to funcoids and other morphisms.

Before starting to discuss topology, author Victor Proton lays the groundwork by investigating the properties of co-Brouwerian lattices and of filters. This foundational knowledge equips the reader with a deeper understanding of the subsequent topological concepts presented.