# 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 be a normal space and a closed subset of it. Then any continuous function can be extended to a continuous function , i.e. one can find a continuous function such that for all . Moreover, if is bounded, then there exists an extension such that

The Urysohn–Brouwer lemma was proved by L.E.J. Brouwer and H. Lebesgue for , by H. Tietze for an arbitrary metric space , 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) |

**How to Cite This Entry:**

Urysohn-Brouwer lemma.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Urysohn-Brouwer_lemma&oldid=23095