Urysohn-Brouwer lemma

Urysohn–Brouwer–Tietze lemma

An assertion on the possibility of extending a continuous function from a subspace of a topological space to the whole space. Let $ X $ be a normal space and $ F $ a closed subset of it. Then any continuous function $ f : F \rightarrow \mathbf R $ can be extended to a continuous function $ g : X \rightarrow \mathbf R $, i.e. one can find a continuous function $ g $ such that $ g ( x) = f ( x) $ for all $ x \in F $. Moreover, if $ f $ is bounded, then there exists an extension $ g $ such that

$$ \sup _ {x \in F } \ | f ( x) | = \sup _ {x \in X } | g ( x) | . $$

The Urysohn–Brouwer lemma was proved by L.E.J. Brouwer and H. Lebesgue for $ X = \mathbf R ^ {n} $, by H. Tietze for an arbitrary metric space $ X $, and by P.S. Urysohn in the above formulation (which may be used as a characterization of normal spaces and is thus best possible).

References

[1]	P.S. Urysohn, "Ueber die Mächtigkeit der zusammenhängenden Mengen" Math. Ann. , 94 (1925) pp. 262–295

Comments

This assertion is also known as the Tietze–Urysohn extension theorem, or even as the Tietze extension theorem.

References

[a1]	R. Engelking, "General topology" , Heldermann (1989)