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 compactHausdorff spaces are normal. It can be generalized by replacing with for some indexing set any retract of or any normal absolute retract whatsoever.
Variationsedit
If is a metric space, a non-empty subset of and is a Lipschitz continuous function with Lipschitz constant then can be extended to a Lipschitz continuous function with same constant This theorem is also valid for Hölder continuous functions, that is, if is Hölder continuous function with constant less than or equal to then can be extended to a Hölder continuous function with the same constant.[4]
Another variant (in fact, generalization) of Tietze's theorem is due to H.Tong and Z. Ercan:[5] Let be a closed subset of a normal topological space If is an upper semicontinuous function, a lower semicontinuous function, and a continuous function such that for each and for each , then there is a continuous extension of such that for each This theorem is also valid with some additional hypothesis if is replaced by a general locally solid Riesz space.[5]
Dugundji (1951) extends the theorem as follows: If is a metric space, is a locally convex topological vector space, is a closed subset of and is continuous, then it could be extended to a continuous function defined on all of . Moreover, the extension could be chosen such that
See alsoedit
Blumberg theorem – Any real function on R admits a continuous restriction on a dense subset of R
^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.
^ abZafer, Ercan (1997). "Extension and Separation of Vector Valued Functions" (PDF). Turkish Journal of Mathematics. 21 (4): 423–430.
Bonan, Edmond (1971), "Relèvements-Prolongements à valeurs dans les espaces de Fréchet", Comptes Rendus de l'Académie des Sciences, Série I, 272: 714–717.
January 01, 1970
tietze, extension, theorem, topology, also, known, tietze, urysohn, brouwer, extension, theorem, urysohn, brouwer, lemma, states, that, real, valued, continuous, function, closed, subset, normal, topological, space, extended, entire, space, preserving, bounded. 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 Contents 1 Formal statement 2 Proof 3 History 4 Equivalent statements 5 Variations 6 See also 7 References 8 External linksFormal statement editIf X displaystyle X nbsp is a normal space andf A R displaystyle f A to mathbb R nbsp is a continuous map from a closed subset A displaystyle A nbsp of X displaystyle X nbsp into the real numbers R displaystyle mathbb R nbsp carrying the standard topology then there exists a continuous extension of f displaystyle f nbsp to X displaystyle X nbsp that is there exists a map F X R displaystyle F X to mathbb R nbsp continuous on all of X displaystyle X nbsp with F a f a displaystyle F a f a nbsp for all a A displaystyle a in A nbsp Moreover F displaystyle F nbsp may be chosen such that sup f a a A sup F x x X displaystyle sup f a a in A sup F x x in X nbsp that is if f displaystyle f nbsp is bounded then F displaystyle F nbsp may be chosen to be bounded with the same bound as f displaystyle f nbsp Proof editThe function F displaystyle F nbsp is constructed iteratively Firstly we definec 0 sup f a a A E 0 a A f a c 0 3 F 0 a A f a c 0 3 displaystyle begin aligned c 0 amp sup f a a in A E 0 amp a in A f a geq c 0 3 F 0 amp a in A f a leq c 0 3 end aligned nbsp Observe that E 0 displaystyle E 0 nbsp and F 0 displaystyle F 0 nbsp are closed and disjoint subsets of A displaystyle A nbsp By taking a linear combination of the function obtained from the proof of Urysohn s lemma there exists a continuous function g 0 X R displaystyle g 0 X to mathbb R nbsp such that g 0 c 0 3 on E 0 g 0 c 0 3 on F 0 displaystyle begin aligned g 0 amp frac c 0 3 text on E 0 g 0 amp frac c 0 3 text on F 0 end aligned nbsp and furthermore c 0 3 g 0 c 0 3 displaystyle frac c 0 3 leq g 0 leq frac c 0 3 nbsp on X displaystyle X nbsp In particular it follows that g 0 c 0 3 f g 0 2 c 0 3 displaystyle begin aligned g 0 amp leq frac c 0 3 f g 0 amp leq frac 2c 0 3 end aligned nbsp on A displaystyle A nbsp We now use induction to construct a sequence of continuous functions g n n 0 displaystyle g n n 0 infty nbsp such that g n 2 n c 0 3 n 1 f g 0 g n 2 n 1 c 0 3 n 1 displaystyle begin aligned g n amp leq frac 2 n c 0 3 n 1 f g 0 g n amp leq frac 2 n 1 c 0 3 n 1 end aligned nbsp We ve shown that this holds for n 0 displaystyle n 0 nbsp and assume that g 0 g n 1 displaystyle g 0 g n 1 nbsp have been constructed Define c n 1 sup f a g 0 a g n 1 a a A displaystyle c n 1 sup f a g 0 a g n 1 a a in A nbsp and repeat the above argument replacing c 0 displaystyle c 0 nbsp with c n 1 displaystyle c n 1 nbsp and replacing f displaystyle f nbsp with f g 0 g n 1 displaystyle f g 0 g n 1 nbsp Then we find that there exists a continuous function g n X R displaystyle g n X to mathbb R nbsp such that g n c n 1 3 f g 0 g n 2 c n 1 3 displaystyle begin aligned g n amp leq frac c n 1 3 f g 0 g n amp leq frac 2c n 1 3 end aligned nbsp By the inductive hypothesis c n 1 2 n c 0 3 n displaystyle c n 1 leq 2 n c 0 3 n nbsp hence we obtain the required identities and the induction is complete Now we define a continuous function F n X R displaystyle F n X to mathbb R nbsp as F n g 0 g n displaystyle F n g 0 g n nbsp Given n m displaystyle n geq m nbsp F n F m g m 1 g n 2 3 m 1 2 3 n c 0 3 2 3 m 1 c 0 displaystyle begin aligned F n F m amp g m 1 g n amp leq left left frac 2 3 right m 1 left frac 2 3 right n right frac c 0 3 amp leq left frac 2 3 right m 1 c 0 end aligned nbsp Therefore the sequence F n n 0 displaystyle F n n 0 infty nbsp is Cauchy Since the space of continuous functions on X displaystyle X nbsp together with the sup norm is a complete metric space it follows that there exists a continuous function F X R displaystyle F X to mathbb R nbsp such that F n displaystyle F n nbsp converges uniformly to F displaystyle F nbsp Since f F n 2 n c 0 3 n 1 displaystyle f F n leq frac 2 n c 0 3 n 1 nbsp on A displaystyle A nbsp it follows that F f displaystyle F f nbsp on A displaystyle A nbsp Finally we observe that F n n 0 g n c 0 displaystyle F n leq sum n 0 infty g n leq c 0 nbsp hence F displaystyle F nbsp is bounded and has the same bound as f displaystyle f nbsp displaystyle square nbsp History editL E J Brouwer and Henri Lebesgue proved a special case of the theorem when X displaystyle X nbsp 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 editThis 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 R displaystyle mathbb R nbsp with R J displaystyle mathbb R J nbsp for some indexing set J displaystyle J nbsp any retract of R J displaystyle mathbb R J nbsp or any normal absolute retract whatsoever Variations editIf X displaystyle X nbsp is a metric space A displaystyle A nbsp a non empty subset of X displaystyle X nbsp and f A R displaystyle f A to mathbb R nbsp is a Lipschitz continuous function with Lipschitz constant K displaystyle K nbsp then f displaystyle f nbsp can be extended to a Lipschitz continuous function F X R displaystyle F X to mathbb R nbsp with same constant K displaystyle K nbsp This theorem is also valid for Holder continuous functions that is if f A R displaystyle f A to mathbb R nbsp is Holder continuous function with constant less than or equal to 1 displaystyle 1 nbsp then f displaystyle f nbsp can be extended to a Holder continuous function F X R displaystyle F X to mathbb R nbsp with the same constant 4 Another variant in fact generalization of Tietze s theorem is due to H Tong and Z Ercan 5 Let A displaystyle A nbsp be a closed subset of a normal topological space X displaystyle X nbsp If f X R displaystyle f X to mathbb R nbsp is an upper semicontinuous function g X R displaystyle g X to mathbb R nbsp a lower semicontinuous function and h A R displaystyle h A to mathbb R nbsp a continuous function such that f x g x displaystyle f x leq g x nbsp for each x X displaystyle x in X nbsp and f a h a g a displaystyle f a leq h a leq g a nbsp for each a A displaystyle a in A nbsp then there is a continuous extension H X R displaystyle H X to mathbb R nbsp of h displaystyle h nbsp such that f x H x g x displaystyle f x leq H x leq g x nbsp for each x X displaystyle x in X nbsp This theorem is also valid with some additional hypothesis if R displaystyle mathbb R nbsp is replaced by a general locally solid Riesz space 5 Dugundji 1951 extends the theorem as follows If X displaystyle X nbsp is a metric space Y displaystyle Y nbsp is a locally convex topological vector space A displaystyle A nbsp is a closed subset of X displaystyle X nbsp and f A Y displaystyle f A to Y nbsp is continuous then it could be extended to a continuous function f displaystyle tilde f nbsp defined on all of X displaystyle X nbsp Moreover the extension could be chosen such that f X conv f A displaystyle tilde f X subseteq text conv f A nbsp See also editBlumberg 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 theoremReferences edit Urysohn Brouwer lemma Encyclopedia of Mathematics EMS Press 2001 1994 Urysohn Brouwer lemma Encyclopedia of Mathematics EMS Press 2001 1994 Urysohn Paul 1925 Uber die Machtigkeit der zusammenhangenden Mengen Mathematische Annalen 94 1 262 295 doi 10 1007 BF01208659 hdl 10338 dmlcz 101038 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 a b 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 editWeisstein Eric W Tietze s Extension Theorem From MathWorld Mizar system proof http mizar org version current html tietze html T23 Bonan Edmond 1971 Relevements Prolongements a valeurs dans les espaces de Frechet Comptes Rendus de l Academie des Sciences Serie I 272 714 717 Retrieved from https en wikipedia org w index php title Tietze extension theorem amp oldid 1222529186, wikipedia, wiki, book, books, library,