Book contents
- Frontmatter
- Contents
- Introduction
- Part One Prologue: The foundations of analysis
- Part Two Functions of a real variable
- 3 Convergent sequences
- 4 Infinite series
- 5 The topology of R
- 6 Continuity
- 7 Differentiation
- 8 Integration
- 9 Introduction to Fourier series
- 10 Some applications
- Appendix A Zorn's lemma and the well-ordering principle
- Index
5 - The topology of R
Published online by Cambridge University Press: 05 May 2013
- Frontmatter
- Contents
- Introduction
- Part One Prologue: The foundations of analysis
- Part Two Functions of a real variable
- 3 Convergent sequences
- 4 Infinite series
- 5 The topology of R
- 6 Continuity
- 7 Differentiation
- 8 Integration
- 9 Introduction to Fourier series
- 10 Some applications
- Appendix A Zorn's lemma and the well-ordering principle
- Index
Summary
In this chapter, we consider some particular sorts of subsets of R, and their relation to convergence. This involves many definitions; familiarity will only come with use. We study the ideas that arise here in a more general setting in Volume II.
Closed sets
We begin by considering intervals in R. A subset I of R is an interval if whenever two numbers belong to it, then so do all the intermediate points: that is, if a < c < b and a, b ∈ I then c ∈ I. R is an interval. The empty set and singleton sets are degenerate intervals. Other examples of intervals are the semi-infinite intervals
(−∞,b)= {x ∈ R : x < b}, (−∞,b]= {x ∈ R : x ≤ b},
(a, ∞)= {x ∈ R : a < x}, [a, ∞)= {x ∈ R : a ≤ x},
and the bounded intervals
(a, b)=(b, a)= {x ∈ R : a < x < b},(a, b]=[b, a)= {x ∈ R : a < x ≤ b},
[a, b)=(b, a]= {x ∈ R : a ≤ x < b},[a, b]=[b, a]= {x ∈ R : a ≤ x ≤ b},
where a < b. It is an easy exercise to show that every interval is of one of these forms. The length of a bounded interval is b − a; the length of R and of semi-infinite intervals is +∞.
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- A Course in Mathematical Analysis , pp. 131 - 146Publisher: Cambridge University PressPrint publication year: 2013