Tietze extension theorem

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.

is a continuous map from a closed subset

carrying the standard topology, then there exists a continuous extension of

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

We now use induction to construct a sequence of continuous functions

and repeat the above argument replacing

Then we find that there exists a continuous function

hence we obtain the required identities and the induction is complete.

Now, we define a continuous function

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

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.

or any normal absolute retract whatsoever.

is a Lipschitz continuous function with Lipschitz constant

can be extended to a Lipschitz continuous function

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

is Hölder continuous function with constant less than or equal to

can be extended to a Hölder continuous function

[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

is an upper semicontinuous function,

a lower semicontinuous function, and

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 locally convex topological vector space,