Metric Spaces (Springer Undergraduate Mathematics Series)
Language: English
Pages: 304
ISBN: 1846283698
Format: PDF / Kindle (mobi) / ePub
The abstract concepts of metric spaces are often perceived as difficult. This book offers a unique approach to the subject which gives readers the advantage of a new perspective on ideas familiar from the analysis of a real line. Rather than passing quickly from the definition of a metric to the more abstract concepts of convergence and continuity, the author takes the concrete notion of distance as far as possible, illustrating the text with examples and naturally arising questions. Attention to detail at this stage is designed to prepare the reader to understand the more abstract ideas with relative ease.
Elementary Real and Complex Analysis
Calculus: Concepts and Methods (2nd Edition)
. . . . . . . . . . . . 200 11.9 Polygonal Connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 12. Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 12.1 Compact Metric Spaces . . . . . . . . . .
r ∈ R+ , the ball X [w ; r) includes a tail of (xn ), and, since every term of (xn ) is in Z, X [w ; r) ∩ Z includes the same tail. If also w ∈ Z, then Z [w ; r) = X [w ; r) ∩ Z (5.3.1) and, since r is arbitrary in R+ , we deduce that (xn ) converges to w in Z also. Theorem 6.4.2 Suppose X is a metric space, w ∈ X and (xn ) is a sequence in X that converges to w. Suppose Y is a metric superspace of X. Then (xn ) converges to w in Y . Proof For each r ∈ R+ , the ball includes a tail of (xn ). Y
each of those parts is continuous; in other words, each restriction of f is continuous at every point of the appropriate constituent part. Does it follow that f is continuous at every point of its domain and is therefore a continuous function? It is important to know that it does not, even if the constituent parts are closed in the domain and mutually disjoint (8.3.7). However, 8.7.1 gives a sufficient condition for the truth of the implication. Theorem 8.7.1 Suppose (X, d) and (Y, e) are metric
of Tororo, Uganda, to whom the book is dedicated. He fostered in me at an early age an appreciation of the beauty and precision of the art of mathematics. I thank Springer’s anonymous reviewers and my fellow mathematicians Christopher Boyd, Thomas J. Laffey, Stefan de Wannemacker, Thomas Unger, Richard Moloney, Robin Harte, J. Brendan Quigley, Remo H¨ ugli, Miriam Logan xvi To the Reader and Patrick Green, who read drafts, commented on the text, alerted me to errors and suggested improvements.
is not a virtual point of X, there exists z ∈ X such that d(w, z) = 0. Then w = z ∈ X. Since w is arbitrary in ∂Y X, it follows that X is closed in Y . So (i) implies (ii). That (ii), (iii) and (iv) are equivalent has already been proved (6.11.3, 4.7.2), and it is clear that (iv) implies (v). Now we suppose that X satisfies (v) and that u is a pointlike function on X that satisfies inf u(X) = 0. For each n ∈ N , An = u−1 ([0 , 1/n]) is non-empty and An+1 ⊆ An . Since u is continuous (10.1.2) and
