Tietze extension theorem

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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.

Contents

Formal statement

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

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

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

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

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

  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. 1 2 Zafer, Ercan (1997). "Extension and Separation of Vector Valued Functions" (PDF). Turkish Journal of Mathematics. 21 (4): 423–430.