topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ho ...

, the Tietze extension theorem (also known as the Tietze–Urysohn–Brouwer extension theorem) states that continuous functions on a

closed subset In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a c ...

of a normal

topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...

can be extended to the entire space, preserving boundedness if necessary.

Formal statement

X

is a

normal space In topology and related branches of mathematics, a normal space is a topological space ''X'' that satisfies Axiom T4: every two disjoint closed sets of ''X'' have disjoint open neighborhoods. A normal Hausdorff space is also called a T4 space ...

and

f : A \to \R

is a continuous map from a

A

X

into the

real number In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...

\R

carrying the

standard topology In mathematics, the real coordinate space of dimension , denoted ( ) or is the set of the -tuples of real numbers, that is the set of all sequences of real numbers. With component-wise addition and scalar multiplication, it is a real vecto ...

, then there exists a of

f

X;

that is, there exists a map

F : X \to \R

continuous on all of

X

with

F(a) = f(a)

for all

a \in A.

Moreover,

F

may be chosen such that

\sup \ ~=~ \sup \,

that is, if

f

is bounded then

F

may be chosen to be bounded (with the same bound as

f

History

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

X

is a finite-dimensional real

vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...

. Heinrich Tietze extended it to all

metric space In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general sett ...

s, and

Pavel Urysohn Pavel Samuilovich Urysohn () (February 3, 1898 – August 17, 1924) was a Soviet mathematician who is best known for his contributions in dimension theory, and for developing Urysohn's metrization theorem and Urysohn's lemma, both of which are ...

proved the theorem as stated here, for normal topological spaces.

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

s and all

compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...

Hausdorff space In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the many ...

s are normal. It can be generalized by replacing

\R

with

\R^J

for some indexing set

J,

any retract of

\R^J,

or any normal

absolute retract In topology, a branch of mathematics, a retraction is a continuous mapping from a topological space into a subspace that preserves the position of all points in that subspace. The subspace is then called a retract of the original space. A defor ...

whatsoever.

Variations

X

is a metric space,

A

a non-empty subset of

X

and

f : A \to \R

is a

Lipschitz continuous In mathematical analysis, Lipschitz continuity, named after German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there ...

function with Lipschitz constant

K,

then

f

can be extended to a Lipschitz continuous function

F : X \to \R

with same constant

K.

This theorem is also valid for Hölder continuous functions, that is, if

f : A \to \R

is Hölder continuous function with constant less than or equal to

1,

then

f

can be extended to a Hölder continuous function

F : X \to \R

with the same constant. Another variant (in fact, generalization) of Tietze's theorem is due to H.Tong and Z. Ercan: Let

A

be a closed subset of a normal topological space

X.

f : X \to \R

is an upper semicontinuous function,

g : X \to \R

a lower semicontinuous function, and

h : A \to \R

a continuous function such that

f(x) \leq g(x)

for each

x \in X

and

f(a) \leq h(a) \leq g(a)

for each

a \in A

, then there is a continuous extension

H : X \to \R

h

such that

f(x) \leq H(x) \leq g(x)

for each

x \in X.

This theorem is also valid with some additional hypothesis if

\R

is replaced by a general locally solid Riesz space.

References

External links

* Weisstein, Eric W.
Tietze's Extension Theorem.
From

MathWorld ''MathWorld'' is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by and licensed to Wolfram Research, Inc. and was partially funded by the National Science Foundation's National Science ...

* Mizar system proof: http://mizar.org/version/current/html/tietze.html#T23 * {{citation , first =Edmond, last =Bonan, title =Relèvements-Prolongements à valeurs dans les espaces de Fréchet, journal = Comptes Rendus de l'Académie des Sciences, Série I, volume =272, year =1971 , pages = 714–717. Theory of continuous functions Theorems in topology

Formal statement

History

Equivalent statements

Variations

See also

References

External links