Topological Spaces: From Distance to Neighborhood

ISBN-13: 9780387949949 / Angielski / Twarda / 1997 / 313 str.

Gerard Buskes; A. Van Rooij; Arnoud Van Rooij

Opis

This book is a text, not a reference, on Point-set Thpology. 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. Th most beginners, Thpology 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. Th 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. Thpological 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 Thpology preeminently is a subject with an extensive ar- ray 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. Th 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, com- plete normality, and extremal disconnectedness-just to mention three terms that are not really esoteric.

Spis treści

I The Line And The Plane.- 1 What Topology Is About.- Topological Equivalence.- Continuity and Convergence.- A Few Conventions.- Extra: Topological Diversions.- Exercises.- 2 Axioms for ?.- Extra: Axiom Systems.- Exercises.- 3 Convergent Sequences and Continuity.- Subsequences.- Uniform Continuity.- The Plane.- Extra: Bolzano (1781-1848).- Exercises.- 4 Curves in the Plane.- Curves.- Homeomorphic Sets.- Brouwer’s Theorem.- Extra: L.E.J. Brouwer (1881-1966).- II Metric Spaces.- 5 Metrics.- Extra: Camille Jordan (1838-1922).- Exercises.- 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).- Exercises.- 7 Completeness.- Extra: Meager Sets and the Mazur Game.- Exercises.- 8 Uniform Convergence.- Extra: Spaces of Continuous Functions.- Exercises.- 9 Sequential Compactness.- Extra: The p-adic Numbers.- Exercises.- 10 Convergent Nets.- Inadequacy of Sequences.- Convergent Nets.- Extra: Knots.- Exercises.- 11 Transition to Topology.- Generalized Convergence.- Topologies.- Extra: The Emergence of the Professional Mathematician.- Exercises.- III Topological Spaces.- 12 Topological Spaces.- Extra: Map Coloring.- Exercises.- 13 Compactness and the Hausdorff Property.- Compact Spaces.- Hausdorff Spaces.- Extra: Hausdorff and the Measure Problem.- Exercises.- 14 Products and Quotients.- Product Spaces.- Quotient Spaces.- Extra: Surfaces.- Exercises.- 15 The Hahn-Tietze-Tong-Urysohn Theorems.- Urysohn’s Lemma.- Interpolation and Extension.- Extra: Nonstandard Mathematics.- Exercises.- 16 Connectedness.- Connected Spaces.- The Jordan Theorem.- Extra: Continuous Deformation of Curves.- Exercises.- 17 Tychonoff’s Theorem.- Extra: The Axiom of Choice.- Exercises.- IV Postscript.- 18 A Smorgasbord for Further Study.- Countability Conditions.- Separation Conditions.- Compactness Conditions.- Compactifications.- Connectivity Conditions.- Extra: Dates from the History of General Topology.- Exercises.- 19 Countable Sets.- Extra: The Continuum Hypothesis.- A Farewell to the Reader.- Literature.- Index of Symbols.- Index of Terms.