Difference between revisions of "Normal space"
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| − | '''Normal space''' (or '''T<sub>4</sub> space''') is a | + | '''Normal space''' (or '''T<sub>4</sub> space''') is a [[Hausdorff 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. Every subspace of a normal space is a [[completely regular space]]. |
By the [[Urysohn lemma]], any 2 disjoint, closed subsets of a normal space can be seperated by a [[continuous function]]. The converse also hold. | By the [[Urysohn lemma]], any 2 disjoint, closed subsets of a normal space can be seperated by a [[continuous function]]. The converse also hold. | ||
[[category: Topology]] | [[category: Topology]] | ||
Revision as of 02:17, April 7, 2007
Normal space (or T4 space) is a Hausdorff 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. Every subspace of a normal space is a completely regular space.
By the Urysohn lemma, any 2 disjoint, closed subsets of a normal space can be seperated by a continuous function. The converse also hold.