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# S28 - a ∈ B d a ε ⊂ X-f X Deﬁne a sequence x n by x...

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1st December 2004 Munkres § 28 Ex. 28.1 (Morten Poulsen). Let d denote the uniform metric. Choose c (0 , 1]. Let A = { 0 , c } ω [0 , 1] ω . Note that if a and b are distinct points in A then d ( a, b ) = c . For any x X the open ball B d ( x, c/ 3) has diameter less than or equal 2 c/ 3, hence B d ( x, c/ 3) cannot contain more than one point of A . It follows that x is not a limit point of A . Ex. 28.6 (Morten Poulsen). Theorem 1. Let ( X, d ) be a compact metric space. If f : X X is an isometry then f is a homeomorphism. Proof. Clearly any isometry is continuous and injective. If f surjective then f - 1 is also an isometry, hence it suffices to show that f is surjective. Suppose f ( X ) X and let a X - f ( X ). Note that f ( X ) is compact, since X compact, hence f ( X ) closed, since X Hausdorff, i.e. X - f ( X ) is open. Thus there exists ε > 0 such that
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Unformatted text preview: a ∈ B d ( a, ε ) ⊂ X-f ( X ). Deﬁne a sequence ( x n ) by x n = ± a, n = 1 f ( x n ) , n > 1 . If n 6 = m then d ( x n , x m ) ≥ ε : Induction on n ≥ 1. If n = 1 then clearly d ( a, x m ) ≥ ε , since x m ∈ f ( X ). Suppose d ( x n , x m ) ≥ ε for all m 6 = n . If m = 1 then d ( x n +1 , x 1 ) = d ( f ( x n ) , a ) ≥ ε . If m > 1 then d ( x n +1 , x m ) = d ( f ( x n ) , f ( x m-1 )) = d ( x n , x m-1 ) ≥ ε . For any x ∈ X the open ball B d ( x, ε/ 3) has diameter less than or equal to 2 ε/ 3, hence B d ( x, ε/ 3) cannot contain more than one point of A . It follows that x is not a limit point of A . ± 1...
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