International Obfuscated C Code Contest

2014-01-31T07:20:53Z

107.208.218.105: The book "Calculated Bets: Computers, Gambling, and Mathematical Modeling to Win" was published in 2001, not 1991.

In [[topology]], the '''Tietze extension theorem''' (also known as the Tietze–Urysohn–Brouwer extension theorem) states that, if ''X'' is a [[normal topological space]] and
:<math>f: A \to \mathbb{R}</math>
is a [[continuous function (topology)|continuous]] map from a [[closed subset]] ''A'' of ''X'' into the [[real number]]s carrying the standard topology, then there exists a continuous map
:<math>F: X \to \mathbb{R}</math>
with ''F''(''a'') = ''f''(''a'') for all ''a'' in ''A''. Moreover, ''F'' may be chosen such that <math>\sup \{ |f(a)| : a \in A \} = \sup \{ |F(x)| : x \in X \}</math>, i.e., if ''f'' is bounded, ''F'' may be chosen to be bounded (with the same bound as ''f''). ''F'' is called a ''continuous extension'' of ''f''.

This theorem is equivalent to the [[Urysohn's lemma]] (which is also equivalent to the normality of the space) and is widely applicable, since all [[metric space]]s and all [[compact space|compact]] [[Hausdorff space]]s are normal. It can be generalized by replacing '''R''' with '''R'''<sup>''J''</sup> for some indexing set ''J'', any retract of '''R'''<sup>''J''</sup>, or any normal [[Deformation retract#Retract|absolute retract]] whatsoever.

The theorem is due to [[Heinrich Franz Friedrich Tietze]].

==External links==
* {{springer|title=Urysohn-Brouwer lemma|id=p/u095860}}
* [[Eric W. Weisstein|Weisstein, Eric W.]] "[http://mathworld.wolfram.com/TietzesExtensionTheorem.html Tietze's Extension Theorem.]" From [[MathWorld]]
* {{planetmath reference|id=4215|title=Tietze extension theorem}}
* {{planetmath reference|id=5566|title=Proof of Tietze extension theorem}}

[[Category:Continuous mappings]]
[[Category:Theorems in topology]]

{{Topology-stub}}

formulasearchengine - User contributions [en]

International Obfuscated C Code Contest