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CHAPTER 6 The Stone-˘Cech Compactification If X and Y are topological spaces, and h : X → Y is a continuous function such that h is a homeomorphism between X and h(X), then h is called an imbedding of X in the topological space Y . Often, topological qualities of a space X can be more easily discerned by imbedding it in a ‘nice’ space Y and analyzing X as a subspace of Y . Many spaces studied in topology are such that for distinct points x1,x2 one can find disjoint open sets U1 and U2 containing x1 and x2, respectively. Any space that possesses this quality is said to be Hausdorff. If Y is a compact Hausdorff space, and X is a dense subset of Y , then Y is said to be a compactification of X. Since spaces that are compact and Hausdorff are relatively easy to study, compactifications are a valuable tool in the study of topological spaces. If X is a Hausdorff space, it may be the case that for any closed set A and point x ∈ X −A there is a continuous real function f on X such that f(x) = 0 and f(a) = 1 for every a ∈ A. Any space X with with this property is said to be completely regular. Every completely regular space is Hausdorff, and many spaces frequently encountered in analysis and topology are completely regular. Definition. A space X is said to be locally compact if, for every point x ∈ X, one can find an open subset U of X containing x such that U has a compact closure. If X is a completely regular space, one can construct a compactification of X by adding one point, called ∞, to X. In fact, this can be done if X is locally compact Hausdorff, but any completely regular space is locally compact Hausdorff. This is, in a sense, the smallest compactification of X, in that we add only a single point to obtain a compact Hausdorff space containing X. There are other ways to compactify a space, but one begins to wonder 34
6. THE STONE- ˘ CECH COMPACTIFICATION 35 if there is a ‘biggest’ compactification of the completely regular space X. It turns out that such a compactification exists, and it was discovered by Marshall Stone and Eduard ˘ Cech. Theorem (Stone- ˘ Cech Compactification 1 ). Let X be completely regular. Then there exists a unique compactification β(X) of X such that (1) Every continuous map of X into a compact Hausdorff space Y extends uniquely to a continuous mapping of β(X) → Y , and (2) Every compactification Y of X is equivalent to a quotient space of β(X). The proof of this theorem is quite large, and will be shown by a sequence of lemmas. We will introduce the theory of quotient spaces when it is needed. Definition. Two compactifications Y1 and Y2 of a space X are said to be equivalent if there is a homeomorphism from Y1 to Y2 that fixes X. Lemma (1). Let X be a space, and let h be an imbedding of X into the compact Hausdorff space Z. Then h induces a unique compactification Y of X, such that there is an imbedding H of Y into Z which agrees with h on X. Proof. Given h, let X0 = h(X) ⊆ Z, and let Y0 be the closure of h(X) in Z. Then Y0 is a compact Hausdorff space and equals the closure of X0, so Y0 is a compactification of X0. Choose a set A that corresponds bijectively with Y0 − X0 via the map k: A → Y0 − X0. Let Y = X ∪ A, and define H : Y → Y0 by H(x) = h(x) for x ∈ X, and H(a) = k(a) for a ∈ A. H is bijective because h and k are bijections. Let U ⊂ Y be open in Y if, and only if, H(U) is open in Y0. Then H is automatically a homeomorphism, being a bijective continuous map. X is a subspace of Y because H agrees with the homeomorphism h when restricted to X. Hence, H is an imbedding of Y into Z. Now, suppose Y1, Y2 are two compactifications of X and that H1 : Y1 → Z and H2 : Y2 → Z are imbeddings that agree with h on X. Then H1(X) = H2(X) = h(X) = X0. Furthermore, since H1 and H2 are continuous and Y1 and Y2 are closed, H1(Y1) = H2(Y2) = 1 The statements and proofs of lemmas 1 through 5 are taken almost directly from [8]. An alternate proof for lemma 6 can be found in [5], although the one provided here is much simpler.
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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Page 34 and 35: CHAPTER 5 The Baire Category Theore
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