ROYAL ROAD TO TOPOLOGY, A: CONVERGENCE OF FILTERS: Convergence of Filters - Szymon...

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).
Key Features:

The framework of convergence theory is easier, more powerful and far-reaching than that of general topology, thanks to an appropriate level of abstraction, enabling us to see the things with inhanced clarity

Convergence theory is for topology, what complex numbers are for real numbers

Several themes of convergence theory have been developed by the author and his collaborators

Thus this book would offer the state of the art of the field