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Urysohn-Brouwer lemma

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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%E2%80%93Brouwer_lemma&oldid=23095