ROYAL ROAD TO TOPOLOGY, A: CONVERGENCE OF FILTERS

Topological spaces are a special case of convergence spaces. This textbook introduces topology within a broader context of convergence theory. The title alludes to advantages of the present approach, which is more gratifying than many traditional ones: you travel more comfortably through mathematical landscapes and you see more.

The book is addressed both to those who wish to learn topology and to those who, being already knowledgeable about topology, are curious to review it from a different perspective, which goes well beyond the traditional knowledge.

Usual topics of classic courses of set-theoretic topology are treated at an early stage of the book — from a viewpoint of convergence of filters, but in a rather elementary way. Later on, most of these facts reappear as simple consequences of more advanced aspects of convergence theory.

The mentioned virtues of the approach stem from the fact that the class of convergences is closed under several natural, essential operations, under which the class of topologies is not! Accordingly, convergence theory complements topology like the field of complex numbers algebraically completes the field of real numbers.

Convergence theory is intuitive and operational because of appropriate level of its abstraction, general enough to grasp the underlying laws, but not too much in order not to lose intuitive appeal.

Contents:

• About the Author
• Preface
• Preliminaries
• From Convergence of Sequences to the Concept of Filter
• Convergence of Filters
• Continuity
• Families of Sets
• Pretopologies
• Topological Structures
• Adherences, Covers, and Compactness
• Topological Concepts
• Functional Study of Topologies
• Functional Partitions and Metrization
• Compact Topologies
• Connected and Disconnected Topologies
• Extensions and Compactifications
• Uniform Structures
• Sequentially Founded Convergences
• Structural Aspects
• Fundamental Classes
• Diagonality and Regularity
• Compactness
• Mixed Properties
• Implementations and Refinements
• Completeness
• Spaces of Maps
• Duality
• Modified Duality
• Outline of Principles
• Category Theoretic Perspective
• Postface
• Bibliography
• List of Symbols
• Index

Readership: Graduate students of mathematics, Academia (topology, analysis).