|
|
| Line 1: |
Line 1: |
| 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
| | Hello, I'm Larae, a 30 year old from Ammerbuch, Germany.<br>My hobbies include (but are not limited to) Amateur radio, Record collecting and watching Two and a Half Men.<br><br>Also visit my website - [http://djr.com/wp-michael.php cheap michael kors] |
| :<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}}
| |
Hello, I'm Larae, a 30 year old from Ammerbuch, Germany.
My hobbies include (but are not limited to) Amateur radio, Record collecting and watching Two and a Half Men.
Also visit my website - cheap michael kors