Editing Tietze extension theorem

{{Short description|Continuous maps on a closed subset of a normal space can be extended}}
[[File:Urysohn.jpeg|thumb | 220x124px | right | Pavel Urysohn]]
In [[topology]], the '''[[Heinrich Tietze|Tietze]] extension theorem''' (also known as the '''Tietze–[[Pavel Urysohn|Urysohn]]–[[L. E. J. Brouwer|Brouwer]] extension theorem''' or '''Urysohn-Brouwer lemma'''<ref name="encyclopediaofmath">{{springer|title=Urysohn-Brouwer lemma|id=p/u095860}}</ref>) states that any [[Real-valued function|real-valued]], [[Continuous function (topology)|continuous function]] on a [[closed subset]] of a [[Normal space|normal]] [[topological space]] can be extended to the entire space, preserving boundedness if necessary.

==Formal statement==
If <math>X</math> is a [[normal space]] and
<math display=block>f : A \to \R</math>
is a [[Continuous function (topology)|continuous]] map from a [[closed subset]] <math>A</math> of <math>X</math> into the [[real number]]s <math>\R</math> carrying the [[Euclidean topology|standard topology]], then there exists a {{em|[[continuous extension]]}} of <math>f</math> to <math>X;</math> that is, there exists a map
<math display=block>F : X \to \R</math>
continuous on all of <math>X</math> with <math>F(a) = f(a)</math> for all <math>a \in A.</math> Moreover, <math>F</math> may be chosen such that 
<math display=block>\sup \{|f(a)| : a \in A\} ~=~ \sup \{|F(x)| : x \in X\},</math>
that is, if <math>f</math> is bounded then <math>F</math> may be chosen to be bounded (with the same bound as <math>f</math>).

==Proof==
The function <math>F</math> is constructed iteratively. Firstly, we define
<math display=block>
\begin{align}
c_0 &= \sup \{|f(a)|:a\in A\}\\
E_0 &= \{a\in A:f(a)\geq c_0/3\}\\
F_0 &=\{a\in A:f(a)\leq -c_0/3\}.
\end{align}
</math>
Observe that <math>E_0</math> and <math>F_0</math> are [[closed set | closed]] and [[disjoint sets | disjoint]] subsets of <math>A</math>. By taking a linear combination of the function obtained from the proof of [[Urysohn's lemma]], there exists a [[continuous function]]  <math>g_0:X\to \mathbb{R}</math> such that
<math display=block>
\begin{align}
	g_0 &= \frac{c_0}{3}\text{ on }E_0\\
	g_0 &= -\frac{c_0}{3}\text{ on }F_0
\end{align}
</math>
and furthermore 
<math display=block>-\frac{c_0}{3}\leq g_0 \leq \frac{c_0}{3}</math>
on <math>X</math>. In particular, it follows that 
<math display=block>
\begin{align}
	|g_0| &\leq \frac{c_0}{3}\\
	|f-g_0| &\leq \frac{2c_0}{3}
\end{align}</math>
on <math>A</math>. We now use [[mathematical induction | induction]] to construct a sequence of continuous functions <math>(g_n)_{n=0}^\infty</math> such that 
<math display=block>
\begin{align}
|g_n|&\leq \frac{2^nc_0}{3^{n+1}}\\
|f-g_0-...-g_{n}|&\leq \frac{2^{n+1}c_0}{3^{n+1}}.
\end{align}</math>
We've shown that this holds for <math>n=0</math> and assume that <math>g_0,...,g_{n-1}</math> have been constructed. Define
<math display=block>c_{n-1} = \sup\{|f(a)-g_0(a)-...-g_{n-1}(a)|:a\in A\}</math>
and repeat the above argument replacing <math>c_0</math> with <math>c_{n-1}</math> and replacing <math>f</math> with <math>f-g_0-...-g_{n-1}</math>. Then we find that there exists a continuous function <math>g_n:X\to \mathbb{R}</math> such that 
<math display=block>
\begin{align}
|g_n|&\leq \frac{c_{n-1}}{3}\\
|f-g_0-...-g_n|&\leq \frac{2c_{n-1}}{3}.
\end{align}
</math>
By the inductive hypothesis, <math>c_{n-1}\leq 2^nc_0/3^n</math> hence we obtain the required identities and the induction is complete. Now, we define a continuous function <math>F_n:X\to \mathbb{R}</math> as 
<math display=block>F_n = g_0+...+g_n.</math>
Given <math>n\geq m</math>,
<math display=block> 
\begin{align}
|F_n - F_m| &= |g_{m+1}+...+g_n|\\
&\leq \left(\left(\frac{2}{3}\right)^{m+1}+...+\left(\frac{2}{3}\right)^{n}\right)\frac{c_0}{3}\\
&\leq \left(\frac{2}{3}\right)^{m+1}c_0.
\end{align}
</math>
Therefore, the sequence <math>(F_n)_{n=0}^\infty</math> is [[Cauchy sequence | Cauchy]]. Since the [[metric space | space]] of continuous functions on <math>X</math> together with the [[uniform norm | sup norm]] is a [[complete metric space]], it follows that there exists a continuous function <math>F:X\to \mathbb{R}</math> such that <math>F_n</math> [[uniform convergence | converges uniformly]] to <math>F</math>. Since 
<math display=block>|f-F_n|\leq \frac{2^{n}c_0}{3^{n+1}}</math>
on <math>A</math>, it follows that <math>F=f</math> on <math>A</math>. Finally, we observe that 
<math display=block> 
|F_n|\leq \sum_{n=0}^\infty |g_n|\leq c_0
</math>
hence <math>F</math> is bounded and has the same bound as <math>f</math>. <math>\square</math>

==History==
[[L. E. J. Brouwer]] and [[Henri Lebesgue]] proved a special case of the theorem, when <math>X</math> is a finite-dimensional real [[vector space]]. [[Heinrich Tietze]] extended it to all [[metric space]]s, and [[Pavel Samuilovich Urysohn|Pavel Urysohn]] proved the theorem as stated here, for normal topological spaces.<ref>{{springer|title=Urysohn-Brouwer lemma|id=p/u095860}}</ref><ref>{{citation|first=Paul|last=Urysohn|author-link=Pavel Samuilovich Urysohn|journal=[[Mathematische Annalen]]|year=1925|volume=94|issue=1|pages=262–295|title=Über die Mächtigkeit der zusammenhängenden Mengen|doi=10.1007/BF01208659|hdl=10338.dmlcz/101038|hdl-access=free}}.</ref>

==Equivalent statements==
This theorem is equivalent to [[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 <math>\R</math> with <math>\R^J</math> for some indexing set <math>J,</math> any retract of <math>\R^J,</math> or any normal [[Deformation retract#Retract|absolute retract]] whatsoever.

==Variations==
If <math>X</math> is a metric space, <math>A</math> a non-empty subset of <math>X</math> and <math>f : A \to \R</math> is a [[Lipschitz continuous]] function with Lipschitz constant <math>K,</math> then <math>f</math> can be extended to a Lipschitz continuous function <math>F : X \to \R</math> with same constant <math>K.</math>
This theorem is also valid for [[Hölder condition|Hölder continuous functions]], that is, if <math>f : A \to \R</math> is Hölder continuous function with constant less than or equal to <math>1,</math> then <math>f</math> can be extended to a Hölder continuous function <math>F : X \to \R</math> with the same constant.<ref>{{cite journal|last1=McShane|first1=E. J.|title=Extension of range of functions|journal=Bulletin of the American Mathematical Society|date=1 December 1934|volume=40|issue=12|pages=837–843|doi=10.1090/S0002-9904-1934-05978-0|doi-access=free}}</ref>

Another variant (in fact, generalization) of Tietze's theorem is due to H.Tong and Z. Ercan:<ref name="Zaf:97">{{cite journal|last1=Zafer|first1=Ercan|title=Extension and Separation of Vector Valued Functions|journal=Turkish Journal of Mathematics|date=1997|volume=21|issue=4|pages=423–430|url=http://journals.tubitak.gov.tr/math/issues/mat-97-21-4/mat-21-4-4-e2104-04.pdf}}</ref>
Let <math>A</math> be a closed subset of a normal topological space <math>X.</math> If <math>f : X \to \R</math> is an [[upper semicontinuous]] function, <math>g : X \to \R</math> a [[lower semicontinuous]] function, and <math>h : A \to \R</math> a continuous function such that <math>f(x) \leq g(x)</math> for each <math>x \in X</math> and <math>f(a) \leq h(a) \leq g(a)</math> for each <math>a \in A</math>, then there is a continuous
extension <math>H : X \to \R</math> of <math>h</math> such that <math>f(x) \leq H(x) \leq g(x)</math> for each <math>x \in X.</math> 
This theorem is also valid with some additional hypothesis if <math>\R</math> is replaced by a general locally solid [[Riesz space]].<ref name="Zaf:97" />

Dugundji (1951) extends the theorem as follows: If <math>X</math> is a metric space, <math>Y</math> is a [[locally convex topological vector space]], <math>A</math> is a closed subset of <math>X</math> and <math>f:A\to Y</math> is continuous, then it could be extended to a continuous function <math>\tilde f</math> defined on all of <math>X</math>. Moreover, the extension could be chosen such that <math>\tilde f(X)\subseteq \text{conv} f(A)</math>

==See also==

* {{annotated link|Blumberg theorem}}
* {{annotated link|Hahn–Banach theorem}}
* {{annotated link|Whitney extension theorem}}

==References==
{{reflist}}

* {{Munkres Topology|edition=2}} <!-- {{sfn|Munkres|2000|p=}} -->

==External links==
* [[Eric W. Weisstein|Weisstein, Eric W.]]  "[http://mathworld.wolfram.com/TietzesExtensionTheorem.html Tietze's Extension Theorem.]" From [[MathWorld]]
* [[Mizar system]] proof: http://mizar.org/version/current/html/tietze.html#T23
* {{citation | first =Edmond| last =Bonan|  title =Relèvements-Prolongements à valeurs dans les espaces de Fréchet| journal = Comptes Rendus de l'Académie des Sciences, Série I|volume =272| year =1971  | pages = 714–717}}.

[[Category:Theory of continuous functions]]
[[Category:Theorems in topology]]