Tietze extension theorem

In topology, the Tietze extension theorem (also known as the Tietze–Urysohn–Brouwer extension theorem or Urysohn-Brouwer lemma^[1]) 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.

Pavel Urysohn

Formal statement

If ${\displaystyle X}$ is a normal space and $${\displaystyle f:A\to \mathbb {R} }$$ is a continuous map from a closed subset ${\displaystyle A}$ of ${\displaystyle X}$ into the real numbers ${\displaystyle \mathbb {R} }$ carrying the standard topology, then there exists a continuous extension of ${\displaystyle f}$ to ${\displaystyle X;}$ that is, there exists a map $${\displaystyle F:X\to \mathbb {R} }$$ continuous on all of ${\displaystyle X}$ with ${\displaystyle F(a)=f(a)}$ for all ${\displaystyle a\in A.}$ Moreover, ${\displaystyle F}$ may be chosen such that $${\displaystyle \sup\{|f(a)|:a\in A\}~=~\sup\{|F(x)|:x\in X\},}$$ that is, if ${\displaystyle f}$ is bounded then ${\displaystyle F}$ may be chosen to be bounded (with the same bound as ${\displaystyle f}$).

Proof

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

History

L. E. J. Brouwer and Henri Lebesgue proved a special case of the theorem, when ${\displaystyle X}$ 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.^[2][3]

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 spaces and all compact Hausdorff spaces are normal. It can be generalized by replacing ${\displaystyle \mathbb {R} }$ with ${\displaystyle \mathbb {R} ^{J}}$ for some indexing set ${\displaystyle J,}$ any retract of ${\displaystyle \mathbb {R} ^{J},}$ or any normal absolute retract whatsoever.

Variations

If ${\displaystyle X}$ is a metric space, ${\displaystyle A}$ a non-empty subset of ${\displaystyle X}$ and ${\displaystyle f:A\to \mathbb {R} }$ is a Lipschitz continuous function with Lipschitz constant ${\displaystyle K,}$ then ${\displaystyle f}$ can be extended to a Lipschitz continuous function ${\displaystyle F:X\to \mathbb {R} }$ with same constant ${\displaystyle K.}$ This theorem is also valid for Hölder continuous functions, that is, if ${\displaystyle f:A\to \mathbb {R} }$ is Hölder continuous function with constant less than or equal to ${\displaystyle 1,}$ then ${\displaystyle f}$ can be extended to a Hölder continuous function ${\displaystyle F:X\to \mathbb {R} }$ with the same constant.^[4]

Another variant (in fact, generalization) of Tietze's theorem is due to H.Tong and Z. Ercan:^[5] Let ${\displaystyle A}$ be a closed subset of a normal topological space ${\displaystyle X.}$ If ${\displaystyle f:X\to \mathbb {R} }$ is an upper semicontinuous function, ${\displaystyle g:X\to \mathbb {R} }$ a lower semicontinuous function, and ${\displaystyle h:A\to \mathbb {R} }$ a continuous function such that ${\displaystyle f(x)\leq g(x)}$ for each ${\displaystyle x\in X}$ and ${\displaystyle f(a)\leq h(a)\leq g(a)}$ for each ${\displaystyle a\in A}$, then there is a continuous extension ${\displaystyle H:X\to \mathbb {R} }$ of ${\displaystyle h}$ such that ${\displaystyle f(x)\leq H(x)\leq g(x)}$ for each ${\displaystyle x\in X.}$ This theorem is also valid with some additional hypothesis if ${\displaystyle \mathbb {R} }$ is replaced by a general locally solid Riesz space.^[5]

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

See also

• Blumberg theorem – Any real function on R admits a continuous restriction on a dense subset of R
• Hahn–Banach theorem – Theorem on extension of bounded linear functionals
• Whitney extension theorem – Partial converse of Taylor's theorem

References

1. "Urysohn-Brouwer lemma", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
2. "Urysohn-Brouwer lemma", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
3. Urysohn, Paul (1925), "Über die Mächtigkeit der zusammenhängenden Mengen", Mathematische Annalen, 94 (1): 262–295, doi:10.1007/BF01208659, hdl:10338.dmlcz/101038.
4. McShane, E. J. (1 December 1934). "Extension of range of functions". Bulletin of the American Mathematical Society. 40 (12): 837–843. doi:10.1090/S0002-9904-1934-05978-0.
5. Zafer, Ercan (1997). "Extension and Separation of Vector Valued Functions" (PDF). Turkish Journal of Mathematics. 21 (4): 423–430.

• Munkres, James R. (2000). Topology (Second ed.). Upper Saddle River, NJ: Prentice Hall, Inc. ISBN 978-0-13-181629-9. OCLC 42683260.

External links

• Weisstein, Eric W. "Tietze's Extension Theorem." From MathWorld
• Mizar system proof: http://mizar.org/version/current/html/tietze.html#T23
• Bonan, Edmond (1971), "Relèvements-Prolongements à valeurs dans les espaces de Fréchet", Comptes Rendus de l'Académie des Sciences, Série I, 272: 714–717.