In topology, the Tietze extension theorem (also known as the Tietze–Urysohn–Brouwer 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.
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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(a) = f(a) for all a in A. Moreover, F may be chosen such that
History
L. E. J. Brouwer 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 (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 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
Another variant (in fact, generalization) of Tietze's theorem is due to Z. Ercan: Let A be a closed subset of a topological space X. If