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$Lecture notes for Math 522 Spring 2012 (Rudin chapter 7)$

$Math 320 - Fall 2010 HW #7 Selected Solutions Problem 5.1.32 Let ...$

6. THE STONE- ˘ CECH COMPACTIFICATION 38 If f : X → Y is a quotient map, define X ∗ = {f −1 (y)| y ∈ Y }. The partition X ∗ of X, is called a quotient space of X. Lemma (6). Every compactification Y of X is equivalent to a quotient space of β(X). Proof. We construct a continuous, surjective, closed map g: β(X) → Y which fixes X. Let j : X → Y be the inclusion map. Since j is continuous and Y is compact Hausdorff, j extends uniquely to a continuous map j ′ on β(X) via Lemma 4. If U is a closed set in β(X) then U is compact in β(X). By the continuity of j ′ it follows that j ′ (U) is compact in Y . By the previous lemma, j ′ (U) is closed in Y . Since Y is the smallest closed set containing X, and X ⊂ j ′ (β(X)), it must be the case that j ′ (β(X)) = Y . We have shown that j ′ is a quotient map. The result follows immediately. This last lemma demonstrates what is meant by saying that the Stone- ˘ Cech compactification is the ‘biggest’ compactification of a completely regular Hausdorff space X; every compactification of X can be mapped bijectively into a quotient space of β(X). The Stone- ˘ Cech compactification is not directly useful in the same way or degree that the theorems of Ascoli-Arzelà, Stone-Weierstrass, Hahn-Banach, and Baire are. Whereas these other theorems primarily provide powerful insights about the structures of a space X, the Stone- ˘ Cech compactification characterized the set of all compactifications of X as a subset of the class of all quotient spaces of β(X).
Bibliography [1] Errett Bishop. A Generalization of the Stone-Weierstrass Theorem. Pacific Journal of Mathematics, 11(3):777–783, 1961. [2] Errett Bishop. Foundations of Constructive Analysis. McGraw-Hill Book Company, Inc., New York, 1967. [3] John B. Conway. Functions of One Complex Variable. Springer-Verlag, New York, second edition, 1978. [4] Bernard Epstein. Linear Functional Analysis: An Introduction to Lebesgue Integration and Infinite Dimensional Problems. W. B. Saunders Company, Philadelphia, 1970. [5] John L. Kelley. General Topology. D. Van Nostrand Company, Inc., Princeton, 1955. [6] A. N. Kolmogorov and S. V . Fomin. Elements of the Theory of Functions and Functional Analysis, volume 1. Graylock Press, Rochester, N.Y., 1957. Trans. Leo F. Boron. [7] Peter Linz. Theoretical Numerical Analysis. Dover Publications, 1979 edition, 2001. [8] James R. Munkres. Topology. Prentice Hall, Upper Saddle River, New Jersey, second edition, 2000. [9] Frigyes Riesz and Bela Sz.-Nagy. Functional Analysis. Frederick Ungar Publishing Co., New York, 1955. Trans. Leo F. Boron. [10] Walter Rudin. Real and Complex Analysis. McGraw-Hill Book Company, New York, 1966. [11] Walter Rudin. Functional Analysis. McGraw-Hill Book Company, New York, 1973. [12] Walter Rudin. Principles of Mathematical Analysis. McGraw-Hill Book Company, New York, third edition, 1976. [13] George F. Simmons. Introduction to Topology and Modern Analysis. McGraw-Hill Book Company, Inc., New York, 1963. 39
Page 1: FIVE MAJOR RESULTS IN ANALYSIS AND
Page 4 and 5: Preface During the analysis of scie
Page 6 and 7: N = {1, 2, 3,...} R = {x| x is a re
Page 8 and 9: (1) x + y ∈ G, (2) (x + y) + z =
Page 10 and 11: TOPOLOGY 6 implies that f is contin
Page 12 and 13: A ⊂ β∈J ∗ TOPOLOGY 8 Uβ, t
Page 14 and 15: TOPOLOGY 10 ensures that f(x0) ∈
Page 16 and 17: 2. THE ASCOLI-ARZELÀ THEOREM 12 Th
Page 18 and 19: 2. THE ASCOLI-ARZELÀ THEOREM 14 ne
Page 20 and 21: that 1 −1 cn(1 − x 2 ) n dx =
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