Tietze extension theorem
In topology, the Tietze extension theorem (also known as the Tietze–Urysohn–Brouwer extension theorem or Urysohn-Brouwer lemma<ref name="encyclopediaofmath">Template:Springer</ref>) states that any real-valued, continuous function on a closed subset of a normal topological space can be extended to the entire space, preserving boundedness if necessary.
Formal statementEdit
If <math>X</math> is a normal space and <math display=block>f : A \to \R</math> is a continuous map from a closed subset <math>A</math> of <math>X</math> into the real numbers <math>\R</math> carrying the standard topology, then there exists a Template:Em 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>).
ProofEdit
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 and 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 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. Since the space of continuous functions on <math>X</math> together with the 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> 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>
HistoryEdit
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 spaces, and Pavel Urysohn proved the theorem as stated here, for normal topological spaces.<ref>Template:Springer</ref><ref>Template:Citation.</ref>
Equivalent statementsEdit
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 spaces and all compact Hausdorff spaces 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 absolute retract whatsoever.
VariationsEdit
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 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>Template:Cite journal</ref>
Another variant (in fact, generalization) of Tietze's theorem is due to H.Tong and Z. Ercan:<ref name="Zaf:97">Template:Cite journal</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>