Statement and Proof of the Tietze Extension Theorem
Statement and Proof of the Tietze Extension Theorem
Statement and Proof of the Tietze Extension Theorem
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PlanetMath: pro<strong>of</strong> <strong>of</strong> <strong>Tietze</strong> extension <strong>the</strong>orem<br />
Remarks: If was a function satisfying , <strong>the</strong>n <strong>the</strong> <strong>the</strong>orem<br />
can be streng<strong>the</strong>ned as follows. Find an extension <strong>of</strong> as above.<br />
The set is closed <strong>and</strong> disjoint from<br />
because for . By Urysohn's lemma <strong>the</strong>re<br />
is a continuous function such that <strong>and</strong> .<br />
Hence is a continuous extension <strong>of</strong> , <strong>and</strong> has <strong>the</strong><br />
property that .<br />
If is unbounded, <strong>the</strong>n <strong>Tietze</strong> extension <strong>the</strong>orem holds as well. To<br />
see that consider . The function has<br />
<strong>the</strong> property that for , <strong>and</strong> so it can be<br />
extended to a continuous function which has <strong>the</strong> property<br />
. Hence is a continuous extension <strong>of</strong> .<br />
"pro<strong>of</strong> <strong>of</strong> <strong>Tietze</strong> extension <strong>the</strong>orem" is owned by bbukh. [ full author list (2) | owner<br />
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Cross-references: unbounded, property, function, sum, side, right, infinity, implies,<br />
converges absolutely, geometric series, converges, sequence, extension, subset,<br />
<strong>the</strong>orem, homeomorphic, Urysohn's lemma, disjoint, continuous function, closed,<br />
topological space, normal, lemma, <strong>Tietze</strong> extension <strong>the</strong>orem<br />
This is version 7 <strong>of</strong> pro<strong>of</strong> <strong>of</strong> <strong>Tietze</strong> extension <strong>the</strong>orem, born on 2004-02-10, modified<br />
2005-05-22.<br />
Object id is 5566, canonical name is <strong>Pro<strong>of</strong></strong>Of<strong>Tietze</strong><strong>Extension</strong><strong>Theorem</strong>2.<br />
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AMS MSC: 54C20 (General topology :: Maps <strong>and</strong> general types <strong>of</strong> spaces defined by maps :: <strong>Extension</strong> <strong>of</strong><br />
maps)<br />
http://planetmath.org/encyclopedia/<strong>Pro<strong>of</strong></strong>Of<strong>Tietze</strong><strong>Extension</strong><strong>Theorem</strong>2.html<br />
9/26/2006