Difference between revisions of "Normal space"
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Normal space is a Hausdorff [[topological space]] in which, given any pair of disjoint closed sets E and F, there exist neighbourhoods U of E and V of F that are disjoint. A product of normal spaces is not necessarily normal, the Sorgenfrey plane is an example of a product of normal spaces that is not normal. On the other hand, every regular space with a countable basis is normal. | Normal space is a Hausdorff [[topological space]] in which, given any pair of disjoint closed sets E and F, there exist neighbourhoods U of E and V of F that are disjoint. A product of normal spaces is not necessarily normal, the Sorgenfrey plane is an example of a product of normal spaces that is not normal. On the other hand, every regular space with a countable basis is normal. | ||
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| + | By the Urysohn lemma, any 2 disjoint, closed subsets of a normal space can be seperated by a contnous function. The converse also hold. | ||
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[[Category:Mathematics]] | [[Category:Mathematics]] | ||
Revision as of 22:07, March 22, 2007
Normal space is a Hausdorff topological space in which, given any pair of disjoint closed sets E and F, there exist neighbourhoods U of E and V of F that are disjoint. A product of normal spaces is not necessarily normal, the Sorgenfrey plane is an example of a product of normal spaces that is not normal. On the other hand, every regular space with a countable basis is normal.
By the Urysohn lemma, any 2 disjoint, closed subsets of a normal space can be seperated by a contnous function. The converse also hold.