Book contents
- Frontmatter
- Contents
- Preface
- PART ONE METRIC SPACES
- 1 Calculus Review
- 2 Countable and Uncountable Sets
- 3 Metrics and Norms
- 4 Open Sets and Closed Sets
- 5 Continuity
- 6 Connectedness
- 7 Completeness
- 8 Compactness
- 9 Category
- PART TWO FUNCTION SPACES
- PART THREE LEBESGUE MEASURE AND INTEGRATION
- References
- Symbol Index
- Topic Index
6 - Connectedness
from PART ONE - METRIC SPACES
Published online by Cambridge University Press: 05 June 2012
- Frontmatter
- Contents
- Preface
- PART ONE METRIC SPACES
- 1 Calculus Review
- 2 Countable and Uncountable Sets
- 3 Metrics and Norms
- 4 Open Sets and Closed Sets
- 5 Continuity
- 6 Connectedness
- 7 Completeness
- 8 Compactness
- 9 Category
- PART TWO FUNCTION SPACES
- PART THREE LEBESGUE MEASURE AND INTEGRATION
- References
- Symbol Index
- Topic Index
Summary
Connected Sets
We have a few details to clean up before we move on to other things; these concern the special role of intervals in ℝ and their use in characterizing the open sets in ℝ given in Chapter Four (see Theorem 4.6 and Exercise 4.25). As we'll see in this section, a better understanding of the special nature of intervals in ℝ will allow us to generalize the intermediate value theorem of calculus. The intermediate value theorem is the formal statement of the informal notion that the graph of a continuous function is “unbroken.” The historical basis of the theorem is the concept of a function as measuring, over time, the relative position of an object moving along a straight line. Thus, if we track the position y = f(x) of a moving object between time x = a and some subsequent time x = b, we would expect the object to “visit” all of the positions y that are intermediate to f(a) and f(b). In short, the continuous image of the time interval [a, b] should contain (at least) the full interval of positions between f(a) and f(b).
The secret here is the intuitively obvious fact that no interval in ℝ can be split into two relatively open parts. Let's prove this by “brute force” for the interval [a, b] (we'll do the other cases shortly).
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- Information
- Real Analysis , pp. 78 - 88Publisher: Cambridge University PressPrint publication year: 2000