Convergence Foundations of Topology

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This textbook is an alternative to a classical introductory book in point-set topology. The approach, however, is radically different from the classical one. It is based on convergence rather than on open and closed sets.

Convergence of filters is a natural generalization of the basic and well-known concept of convergence of sequences, so that convergence theory is more natural and intuitive to many, perhaps most, students than classical topology. On the other hand, the framework of convergence is easier, more powerful and far-reaching which highlights a need for a theory of convergence in various branches of analysis.

Convergence theory for filters is gradually introduced and systematically developed. Topological spaces are presented as a special subclass of convergence spaces of particular interest, but a large part of the material usually developed in a topology textbook is treated in the larger realm of convergence spaces.

Contents:

• Introduction
• Families of Sets
• Convergences
• Continuity
• Pretopologies
• Diagonality and Regularity
• Types of Separation
• Pseudotopologies
• Compactness
• Completeness in Metric Spaces
• Completeness
• Connectedness
• Compactifications
• Classification of Spaces
• Classification of Maps
• Spaces of Maps
• Duality
• Functional Partitions and Metrization
• Set Theory

Readership: Undergraduates, graduates and academia in the fields of analysis and topology.
Key Features:

• The authors are two leading specialists of convergence spaces, with extensive experience that leads them to the right way to teach topology from the “convergence viewpoint”
• The “right level of abstraction” makes presentation and understanding of convergence spaces and topology proofs easier