# Topological Spaces: From Distance to Neighborhood by Gerard Buskes and Arnoud van Rooij

PART I. THE LINE AND THE PLANE

Chapter 1. What Topology Is About
• Topological Equivalence
• Continuity and Convergence
• A Few Conventions
• Extra: Topological Diversions
• Exercises
Chapter 2. Axioms for |R
• Extra: Axiom Systems
• Exercises
Chapter 3. Convergent Sequences and Continuity
• Subsequences
• Uniform Continuity
• The Plane
• Extra: Bolzano (1781-1848)
• Exercises
Chapter 4. Curves in the Plane
• Curves
• Homeomorphic Sets
• Brouwer's Theorem
• Extra: L.E.J. Brouwer (1881-1966)

PART II. METRIC SPACES

Chapter 5. Metrics
• Extra: Camille Jordan (1838-1922)
• Exercises
Chapter 6. Open and Closed Sets
• Subsets of a Metric Space
• Collections of Sets
• Similar Metrics
• Interior and Closure
• The Empty Set
• Extra: Cantor (1845-1918)
Chapter 7. Completeness
• Extra: Meager Sets and the Mazur Game
• Exercises
Chapter 8. Uniform Convergence
• Extra: Spaces of Continuous Functions
• Exercises
Chapter 9. Sequential Compactness
• Exercises
Chapter 10. Convergent Nets
• Convergent Nets
• Extra: Knots
• Exercises
Chapter 11. Transition to Topology
• Generalized Convergence
• Topologies
• Extra: The Emergence of the Professional Mathematician
• Exercises

PART III. TOPOLOGICAL SPACES

Chapter 12. Topological Spaces
• Extra: Map Coloring
• Exercises
Chapter 13. Compactness and the Hausdorff Property
• Compact Spaces
• Hausdorff Spaces
• Extra: Hausdorff and the Measure Problem
• Exercises
Chapter 14. Products and Quotients
• Product Spaces
• Quotient Spaces
• Extra: Surfaces
• Exercises
Chapter 15. The Hahn-Tietze-Tong-Urysohn Theorems
• Urysohn's Lemma
• Interpolation and Extension
• Extra: Nonstandard Mathematics
• Exercises
Chapter 16. Connectedness
• Connected Spaces
• The Jordan Theorem
• Extra: Continuous Deformation of Curves
• Exercises
Chapter 17. Tychonoff's Theorem
• Extra: The Axiom of Choice
• Exercises

PART IV. POSTSCRIPT

Chapter 18. A Smorgasbord for Further Study
• Countability Conditions
• Separation Conditions
• Compactness Conditions
• Compactifications
• Connectivity Conditions
• Extra: Dates from the History of General Topology
• Exercises
Chapter 19. Countable Sets
• Extra: The Continuum Hypothesis

Preface

This book is a text, not a reference, on Point-set Topology. It addresses itself to the student who is proficient in Calculus and has some experience with mathematical rigor, acquired, e.g., via a course in Advanced Calculus or Linear Algebra.

To most beginners, Topology offers a double challenge. In addition to the strangeness of concepts and techniques presented by any new subject, there is an abrupt rise of the level of abstraction. It is a bad idea to teach a student two things at the same moment. To mitigate the culture shock, we move from the special to the general, dividing the book into three parts:

1. The Line and the Plane
2. Metric Spaces
3. Topological Spaces

In this way, the student has ample time to get acquainted with new ideas while still on familiar territory. Only after that, the transition to a more abstract point of view takes place.

Elementary Topology preeminently is a subject with an extensive array of technical terms indicating properties of topological spaces. In the main body of the text, we have purposely restricted our mathematical vocabulary as much as is reasonably possible. Such an enterprise is risky. Doubtlessly, many readers will find us too thrifty. To meet them halfway, in Chapter 18 we briefly introduce and discuss a number of topological properties, but even there we do not touch on paracompactness, complete normality, and extremal disconnectedness - just to mention three terms that are not really esoteric.

In a highly abstract topic like ours, it aids a student to focus on a central theme. The theme of our book is convergence. We show how, for |R^n and for metric spaces in general, concepts such as "continuous" and "closed" can be described in terms of convergent sequences. After that, in any given set X we introduce convergence of nets relative to any given collection w of subsets of X. This convergence leads in a natural way to the notion of a topology. The idea behind this somewhat unconventional approach is threefold.

First, it shows that the definition of "topology" is less artificial than it seems to be. Without this presentation, the definition appears to stem from an arbitrary selection of properties of the system of open sets in |R^n, and it is not clear why precisely these properties are the relevant ones. (The reader who finds this a digression can skip Chapter 11; in Chapter 12, the definition and some basic facts are repeated without the motivation.)

Second, it relegates the notion of "topology" to a place in the second rank. When one studies a topological space, often the topology itself is less relevant than a subbase for it (the collection w in the situation described above.) A case in point is the product topology on a Cartesian product of topological spaces: all that really matters is a subbase, and the fact that this subbase generates a topology is quite immaterial.

Third, convergent nets form a very useful tool in Topology, deserving much more attention than they usually get.

We do not assume previous knowledge of the axiomatic approach. As, however, a rigorous theory of topological spaces must have a firm base in Analysis, we start with a brief axiomatic treatment of the real-number system, explaining what axioms are and what purpose they serve.

We do not assume previous knowledge of Set Theory either. (Indeed, to be on the safe side, we have added a chapter on countability.) On the other hand, Topology unavoidably leads to nontrivial set-theoretic problems. Accordingly, in connection with the Tychonoff Theorem, we pay close attention to the Axiom of Choice and Zorn's Lemma and their role in mathematics.

The pace of this book is relaxed with a gradual acceleration. For instance, the first three chapters and part of Chapter 4 can be relegated to home reading for a well-prepared student. However, the easy initial pace makes the first nine chapters a balanced course in metric spaces for undergraduates. The book contains more than enough material for a two-semester graduate course.

As with all mathematical learning, a substantial amount of practice is indispensable. We offer exercises of varying degrees of difficulty. Some are routine, others illustrate results of the text, and yet others go beyond the text. We have carefully crafted these exercises. Accordingly, one will find many of them, in particular the complicated ones, sectioned into more digestible pieces with hints.

Finally, in most chapters we present an "extra", a brief foray outside Topology. A beginning student is apt to consider each branch of mathematics as an autonomous unit, isolated from the rest, and also to think that mathematics is a museum piece, something created in olden times by our forefathers, that can be seen and even studied, but not touched. Our purpose of the extras is to illustrate the many connections between Topology and other subjects, such as Analysis and Set Theory. Also, in our extras we try to show that Topology was and still is built by individuals, who sometimes made mistakes. We encourage the reader to consider these extras to be part of the course. The extras are extra, not extraneous.