Difference between revisions of "Urysohn lemma"
Jump to navigation
Jump to search
(Elaborating the converse, and adding historical notes.) |
(Urysohn drowned in France in 1924. *LOL*) |
||
| Line 4: | Line 4: | ||
</blockquote> | </blockquote> | ||
The converse of the Urysohn lemma also holds: if there is such a continuous function for any two disjoint closed sets ''A'' and ''B'' in ''X'', then ''X'' is normal. | The converse of the Urysohn lemma also holds: if there is such a continuous function for any two disjoint closed sets ''A'' and ''B'' in ''X'', then ''X'' is normal. | ||
| − | |||
| − | |||
[[category: Topology]] | [[category: Topology]] | ||
Revision as of 20:41, August 26, 2008
The Urysohn lemma states:
If A, B are two disjoint, closed subsets of a normal space X, then there exist a continuous function f : X → [0, 1], such that f(A) = 0 and f(B) = 1.
The converse of the Urysohn lemma also holds: if there is such a continuous function for any two disjoint closed sets A and B in X, then X is normal.