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# S33 - 9th June 2005 Munkres Β 33 Ex 33.1(Morten Poulsen Let...

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Unformatted text preview: 9th June 2005 Munkres Β§ 33 Ex. 33.1 (Morten Poulsen). Let r β [0 , 1]. Recall from the proof of the Urysohn lemma that if p < q then U p β U q . Furthermore, recall that U q = β if q < 0 and U p = X if p > 1. Claim 1. f- 1 ( { r } ) = T p>r U p- S q<r U q , p, q β Q . Proof. By the construction of f : X β [0 , 1], p> U p- [ q< U q = p> U p = f- 1 ( { } ) and p> 1 U p- [ q< 1 U q = X- [ q< 1 U q = f- 1 ( { 1 } ) . Now assume r β (0 , 1). β β β: Let x β f- 1 ( { r } ), i.e. f ( x ) = r = inf { p | x β U p } . Note that x / β S q<r U q , since f ( x ) = r . Suppose there exists t > r , t β Q , such that x / β U t . Since f ( x ) = r , there exists s β Q such that r β€ s < t and x β U s . Now x β U s β U s β U t , contradiction. It follows that x β T p>r U p- S q<r U q . β β β: Let x β T p>r U p- S q<r U q . Note that f ( x ) β€ r , since x β T p>r U p . Suppose f ( x ) < r , i.e. there exists t < r such that x β U t β S q<r U q , contradiction. It follows that, contradiction....
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S33 - 9th June 2005 Munkres Β 33 Ex 33.1(Morten Poulsen Let...

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