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.

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 ).

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 .

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] }

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.

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

- 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

In mathematics, the **Borsuk–Ulam theorem** states that every continuous function from an *n*-sphere into Euclidean *n*-space maps some pair of antipodal points to the same point. Here, two points on a sphere are called antipodal if they are in exactly opposite directions from the sphere's center.

In mathematics, specifically general topology, **compactness** is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space. The idea is that a compact space has no "punctures" or "missing endpoints", i.e., it includes all *limiting values* of points. For example, the open interval (0,1) would not be compact because it excludes the limiting values of 0 and 1, whereas the closed interval [0,1] would be compact. Similarly, the space of rational numbers is not compact, because it has infinitely many "punctures" corresponding to the irrational numbers, and the space of real numbers is not compact either, because it excludes the two limiting values and . However, the *extended* real number line*would* be compact, since it contains both infinities. There are many ways to make this heuristic notion precise. These ways usually agree in a metric space, but may not be equivalent in other topological spaces.

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In functional analysis and related branches of mathematics, the **Banach–Alaoglu theorem** states that the closed unit ball of the dual space of a normed vector space is compact in the weak* topology. A common proof identifies the unit ball with the weak-* topology as a closed subset of a product of compact sets with the product topology. As a consequence of Tychonoff's theorem, this product, and hence the unit ball within, is compact.

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In mathematics, the **Riemann–Lebesgue lemma**, named after Bernhard Riemann and Henri Lebesgue, states that the Fourier transform or Laplace transform of an *L*^{1} function vanishes at infinity. It is of importance in harmonic analysis and asymptotic analysis.

In mathematics, **subharmonic** and **superharmonic** functions are important classes of functions used extensively in partial differential equations, complex analysis and potential theory.

In mathematics, a real or complex-valued function *f* on *d*-dimensional Euclidean space satisfies a **Hölder condition**, or is **Hölder continuous**, when there are real constants *C* ≥ 0, > 0, such that

The **Katětov–Tong insertion theorem** is a theorem of point-set topology proved independently by Miroslav Katětov and Hing Tong in the 1950s. The theorem states the following:

In mathematics, **calculus on Euclidean space** is a generalization of calculus of functions in one or several variables to calculus of functions on Euclidean space as well as a finite-dimensional real vector space. This calculus is also known as **advanced calculus**, especially in the United States. It is similar to multivariable calculus but is somewhat more sophisticated in that it uses linear algebra more extensively and covers some concepts from differential geometry such as differential forms and Stokes' formula in terms of differential forms. This extensive use of linear algebra also allows a natural generalization of multivariable calculus to calculus on Banach spaces or topological vector spaces.

- ↑ "Urysohn-Brouwer lemma",
*Encyclopedia of Mathematics*, EMS Press, 2001 [1994] - ↑ "Urysohn-Brouwer lemma",
*Encyclopedia of Mathematics*, EMS Press, 2001 [1994] - ↑ Urysohn, Paul (1925), "Über die Mächtigkeit der zusammenhängenden 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 . - 1 2 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.

- 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.

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