# 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