Tietze extension theorem


In topology, the Tietze extension theorem states that continuous functions on a closed subset of a normal topological space can be extended to the entire space, preserving boundedness if necessary.

Formal statement

If X is a normal topological space and
is a continuous map from a closed subset A of X into the real numbers carrying the standard topology, then there exists a continuous map
with F = f for all a in A. Moreover, F may be chosen such that, i.e., if f is bounded, F may be chosen to be bounded. F is called a continuous extension of f.

History

and Henri Lebesgue proved a special case of the theorem, when X is a finite-dimensional real vector space. Heinrich Tietze extended it to all metric spaces, and Paul Urysohn proved the theorem as stated here, for normal topological spaces.

Equivalent statements

This theorem is equivalent to Urysohn's lemma and is widely applicable, since all metric spaces and all compact Hausdorff spaces are normal. It can be generalized by replacing R with RJ for some indexing set J, any retract of RJ, or any normal absolute retract whatsoever.

Variations

If X is a metric space, A a non-empty subset of X and is a Lipschitz continuous function with Lipschitz constant K, then f can be extended to a Lipschitz continuous function with same constant K.
This theorem is also valid for Hölder continuous functions, that is, if is Hölder continuous function, f can be extended to a Hölder continuous function with the same constant.
Another variant of Tietze's theorem is due to Z. Ercan:
Let A be a closed subset of a topological space X. If is an upper-semicontinuous function,, is a lower-semicontinuous function, and a continuous function such that f ≤ g for each x in X and f ≤ h ≤ g for each a in A, then there is a continuous
extension of h such that f ≤ H ≤ g 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.