Normal space

From Conservapedia
Jump to: navigation, search

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.