Difference between revisions of "Normal space"
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| − | Normal space is a | + | A '''Normal 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]]. A normal space which is also T<sub>1</sub> is called T<sub>4</sub>. |
| − | [[Category: | + | |
| + | By the [[Urysohn lemma]], any 2 disjoint, closed subsets of a normal space can be separated by a [[continuous function]]. The converse also hold. | ||
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| + | [[Category:Topology]] | ||
Latest revision as of 16:46, July 13, 2016
A Normal 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. A normal space which is also T1 is called T4.
By the Urysohn lemma, any 2 disjoint, closed subsets of a normal space can be separated by a continuous function. The converse also hold.