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6. THE STONE- ˘ CECH COMPACTIFICATION 36 Y0. Then H −1 2 ◦ H1 is a homeomorphism of Y1 with Y2 that fixes X, so that Y1 and Y2 are equivalent. The following lemma is taken from [8], and its proof will be omitted. Lemma (2). Let X be a space in which one-point sets are closed. Suppose that {fα} α∈J is an indexed family of continuous functions fα : X → R such that for each x0 ∈ X and for each open set U of X containing x0, there is an α ∈ J such that fα is positive at x0 and vanishes outside of U. Then F : X → R J , defined by F(x) = (fα(x)) α∈J is an imbedding of X in R J . If fα maps X into [0, 1] for each α, then F is an imbedding of X into [0, 1] J . We construct the Stone- ˘ Cech compactification of X, β(X), as follows: Let {fα} α∈J be the collection of all bounded continuous real functions on X. For each {fα} α∈J , choose a closed interval Iα in R containing fα(X). For instance, let Iα = [inf fα(x), sup fα(x)]. Let x x h : X → Iα be defined by h(x) = (fα(x)) α∈J . By the Tychonoff theorem, which states α∈J that the arbitrary product of compact spaces is compact, α∈J Iα is compact. Because X is completely regular, {fα} separates points from closed sets in X, so that h is an embedding by Lemma 2. Let β(X) be the unique compactification of X induced by the imbedding h guaranteed by Lemma 1. Then there is an imbedding Hβ of β(X) into Iα that agrees with h when restricted to X. Definition. A real function f is said to be bounded if there exists a non-negative real number M such that |f(x)| ≤ M for each x in X. Lemma (3). Let X be a space. Then every bounded continuous map f : X → R extends uniquely to a continuous map of β(X) into R. Proof. If f ∈ {fα} α∈J , then f = fβ for some β. Let πβ : α∈J α∈J Iα → Iβ be defined by πβ ((xα)α∈J) = xβ. πβ is clearly continuous, so that πβ ◦ Hβ : β(X) → Iβ is a continuous extension of f to β(X), since for x ∈ X, πβ (Hβ(x)) = πβ (h(x)) = πβ (fα(x)) α∈J = fβ(x). By Lemma 1, πβ ◦ Hβ is unique. The next lemma proves Part (1) of the theorem.
6. THE STONE- ˘ CECH COMPACTIFICATION 37 Lemma (4). Every continuous map of X into a compact Hausdorff space Y extends uniquely to a continuous mapping of β(X) into Y . Proof. Since Y is completely regular, it can be imbedded in [0, 1] J for some J by Lemma 2, so we can assume that Y ⊂ [0, 1] J . Then each component function fα of f is a bounded continuous real function of X, so by Lemma 3, fα extends uniquely to a continuous map gα of β(X) into R. Let g : β(X) → R J be defined by g(x) = (gα(x)) α∈J . Then g is continuous because R J has the product topology, and from this continuity we note that g(β(X)) = g X ⊂ g(X) = f(X) ⊂ Y = Y , so that g maps β(X) into Y . Lemma 5 shows the uniqueness of the Stone- ˘ Cech compactification. Lemma (5). Any two compactification of X with property (1) are equivalent. Proof. Let β1(X) and β2(X) be two compactifications of X satisfying the extension property (1) of the Stone- ˘ Cech compactification. Let j2: X → β2(X) be the inclusion map. Since j2 is continuous, and because β1(X) has property (1), there is a unique continuous extension f2 of j1 to β1(X). Similarly, the inclusion map j1 : X → β1(X) has a unique continuous extension f1 defined on β2(X). Then f1 ◦ f2 maps β1 into itself and fixes X, so that f1 ◦ f2 is a continuous extension of iX, the identity map of X. Similarly, f2 ◦ f1 is a continuous extension of iX. By Lemma 4, f1 ◦ f2 = iβ1(X), and f2 ◦ f1 = iβ2(X), so that f1 and f2 are homeomorphisms which fix X. Definition. Let X and Y be topological spaces. Then a surjective map f : X → Y is said to be a quotient map if a set U is open in Y if, and only if, f −1 (U) is open in X. Lemma. If X is Hausdorff and A is a compact subset of X, then A is closed. Proof. We will show that X − A is open. Given a point x ∈ X − A, choose, for each a ∈ A, disjoint open sets Ua and Va containing x and a, respectively. Since {Va} a∈A is an open cover of A, there is a finite subcollection of these open sets, say Va1,...,Van covering n A. Then U = is open, x ∈ U, and U ∩ A is empty. i=1 Uai
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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