Urysohn extension theorem
Suppose that is a metric space and is closed in . If are completely separated sets in , then they are contained in disjoint zero sets and in . Since and are closed in , and is closed in , and are closed in . Since is a metric space, and are zero sets in . Since and and are disjoint, and are completely separated in as well. By Urysohn Extension Theorem, any bounded continuous function defined on can be extended to a continuous function on , which is the statement of the metric space version of the Tietze extension theorem.
|Title||Urysohn extension theorem|
|Date of creation||2013-03-22 17:01:43|
|Last modified on||2013-03-22 17:01:43|
|Last modified by||CWoo (3771)|