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.
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.