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Statement and Proof of the Tietze Extension Theorem

Statement and Proof of the Tietze Extension Theorem


PlanetMath: proof of Tietze extension theorem by for and for . Now is continuous and we can extend it to a continuous function . By Urysohn's lemma, is normal because is a continuous function such that for and for . Conversely, let be normal and be closed in . By the lemma, there is a continuous function such that for and for . Since is continuous, the lemma tells us there is a continuous function such that for and for . By repeated application of the lemma we can construct a sequence of continuous functions such that for all . , and for Define . Since and converges as a geometric series, then converges absolutely and uniformly, so is a continuous function defined everywhere. Moreover implies that . Page 2 of 4 Now for , we have that and as goes to infinity, the right side goes to zero and so the sum goes to . Thus Therefore extends . 9/26/2006

PlanetMath: proof of Tietze extension theorem Remarks: If was a function satisfying , then the theorem can be strengthened as follows. Find an extension of as above. The set is closed and disjoint from because for . By Urysohn's lemma there is a continuous function such that and . Hence is a continuous extension of , and has the property that . If is unbounded, then Tietze extension theorem holds as well. To see that consider . The function has the property that for , and so it can be extended to a continuous function which has the property . Hence is a continuous extension of . "proof of Tietze extension theorem" is owned by bbukh. [ full author list (2) | owner history (1) ] View style: This object's parent. (view preamble) Cross-references: unbounded, property, function, sum, side, right, infinity, implies, converges absolutely, geometric series, converges, sequence, extension, subset, theorem, homeomorphic, Urysohn's lemma, disjoint, continuous function, closed, topological space, normal, lemma, Tietze extension theorem This is version 7 of proof of Tietze extension theorem, born on 2004-02-10, modified 2005-05-22. Object id is 5566, canonical name is ProofOfTietzeExtensionTheorem2. Accessed 2054 times total. Classification: HTML with images reload Page 3 of 4 AMS MSC: 54C20 (General topology :: Maps and general types of spaces defined by maps :: Extension of maps) 9/26/2006

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