Tietze extension theorem

In topology, the Tietze extension theorem (also known as the Tietze–UrysohnBrouwer 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

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If   is a normal space and   is a continuous map from a closed subset   of   into the real numbers   carrying the standard topology, then there exists a continuous extension of   to   that is, there exists a map   continuous on all of   with   for all   Moreover,   may be chosen such that   that is, if   is bounded then   may be chosen to be bounded (with the same bound as  ).

Proof

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The function   is constructed iteratively. Firstly, we define   Observe that   and   are closed and disjoint subsets of  . By taking a linear combination of the function obtained from the proof of Urysohn's lemma, there exists a continuous function   such that   and furthermore   on  . In particular, it follows that   on  . We now use induction to construct a sequence of continuous functions   such that   We've shown that this holds for   and assume that   have been constructed. Define   and repeat the above argument replacing   with   and replacing   with  . Then we find that there exists a continuous function   such that   By the inductive hypothesis,   hence we obtain the required identities and the induction is complete. Now, we define a continuous function   as   Given  ,   Therefore, the sequence   is Cauchy. Since the space of continuous functions on   together with the sup norm is a complete metric space, it follows that there exists a continuous function   such that   converges uniformly to  . Since   on  , it follows that   on  . Finally, we observe that   hence   is bounded and has the same bound as  .  

History

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L. E. J. Brouwer and Henri Lebesgue proved a special case of the theorem, when   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

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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   with   for some indexing set   any retract of   or any normal absolute retract whatsoever.

Variations

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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 also

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References

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  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. ^ a b Zafer, Ercan (1997). "Extension and Separation of Vector Valued Functions" (PDF). Turkish Journal of Mathematics. 21 (4): 423–430.
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