General Topology as Ordered Semigroup Actions: Book 1

The reader will learn from this book advanced topics in mathematics. This text consists of a series of simple to advanced topology topics. However, all the competences needed for advanced mathematical studies are extensively discussed and evaluated. General Topology as Ordered Semigroup Actions is perfect for teachers in topology or elective undergraduate and graduate studies. In this work the writer introduces and study in details the concepts of funcoid which generalize proximity spaces and reloids which generalize uniform spaces, and generalizations thereof. The concept of funcoid is generalized concept of proximity; the concept of reloid is cleared from superfluous details (generalized) concept of uniformity. Also funcoids and reloids can be considered as a generalization of (oriented) graphs, this provides us with a common generalization of calculus and discrete mathematics. The concept of continuity is defined by an algebraic formula (instead of old messy epsilon-delta notation) for arbitrary morphisms (including funcoids and reloids) of a partially ordered category. In one formula continuity, proximity continuity, and uniform continuity are generalized. Also the author defines connectedness for funcoids and reloids. Then he considers generalizations of funcoids: pointfree funcoids and generalization of pointfree funcoids: staroids and multifuncoids. Also I define several kinds of products of funcoids and other morphisms. Before going to topology, this book studies property of co-brouwerian lattices and filters as well. This is a great resource which is a must have for you. However, it is not a highly standard book and you will therefore need to back it up with other topology materials. Purchasing this book, you support carbon accounting and DeSci (decentralized science).