Book contents
- Frontmatter
- Contents
- Preface
- Acknowledgements
- To the reader
- Prologue
- Part I Causality and differentiable structure
- Part II Geometrical points and measurement theory
- Mathematical appendices
- A1 Sets and mappings
- A2 The real number system
- A3 Point-set topology
- A4 Completions
- A5 Measure and integral
- A6 Hilbert space, operators and spectral theory
- A7 Conditional expectations
- A8 Fibre bundles, differentiable manifolds, Lie groups and Lie algebras
- List of Symbols for Part I
- References
- Index
A3 - Point-set topology
Published online by Cambridge University Press: 04 August 2010
- Frontmatter
- Contents
- Preface
- Acknowledgements
- To the reader
- Prologue
- Part I Causality and differentiable structure
- Part II Geometrical points and measurement theory
- Mathematical appendices
- A1 Sets and mappings
- A2 The real number system
- A3 Point-set topology
- A4 Completions
- A5 Measure and integral
- A6 Hilbert space, operators and spectral theory
- A7 Conditional expectations
- A8 Fibre bundles, differentiable manifolds, Lie groups and Lie algebras
- List of Symbols for Part I
- References
- Index
Summary
This section provides a thumbnail sketch of those elements of point-set topology (also called general topology or just plain topology) that are used in this book. The subject grew out of attempts to rid the notion of continuity of its traditional dependence on the notion of distance. It turned out that continuity could be defined without using real numbers at all; the subject could be founded, instead, on the calculus of sets. Unfamiliarity with the latter is perhaps the main source of difficulty for the beginner.
Detailed treatments of the material discussed below may be found in standard textbooks such as (Kelley, 1955, Willard, 1970 and Munkres, 1975). Of these, the one by Munkres will perhaps be the easiest for the physicist.
Topological spaces
Point-set topology (usually called topology for short) may be regarded as the study of the notions of convergence of sequences1 and continuity of maps without using the notion of real numbers. In the theory of functions of a real variable, both of these notions are intimately related to that of neighbourhoods of a point. A neighbourhood of a point x on the real line is any subset that contains an open interval (x − a, x + a) around x, where a > 0; usually a is a small number, but it does not have to be.
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- Publisher: Cambridge University PressPrint publication year: 2010