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Statement and Proof of the Tietze Extension Theorem

PlanetMath: pro**of** **of** **Tietze** extension **the**orem 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, **the**re is a continuous function such that for **and** for . Since is continuous, **the** lemma tells us **the**re 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, **the**n 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 . http://planetmath.org/encyclopedia/**Pro of**Of

PlanetMath: pro**of** **of** **Tietze** extension **the**orem Remarks: If was a function satisfying , **the**n **the** **the**orem can be streng**the**ned as follows. Find an extension **of** as above. The set is closed **and** disjoint from because for . By Urysohn's lemma **the**re is a continuous function such that **and** . Hence is a continuous extension **of** , **and** has **the** property that . If is unbounded, **the**n **Tietze** extension **the**orem 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** . "pro**of** **of** **Tietze** extension **the**orem" 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, **the**orem, homeomorphic, Urysohn's lemma, disjoint, continuous function, closed, topological space, normal, lemma, **Tietze** extension **the**orem This is version 7 **of** pro**of** **of** **Tietze** extension **the**orem, born on 2004-02-10, modified 2005-05-22. Object id is 5566, canonical name is **Pro of**Of

- Page 1 and 2: PlanetMath: Tietze extension theore
- Page 3: PlanetMath: proof of Tietze extensi