A topological space in which points are closed sets is called a normal space if every disjoint pair of closed sets E and F have disjoint neighborhoods.

Normal spaces are regular spaces; the converse is not necessarily true. However, it is easy to see that metric spaces are normal.

In mathematics a topological space X is said to be normal iff one-point sets are closed in X and if for every pair of disjoint closed sets A and B there exist disjoint open sets U and V such that A is contained in U and B is contained in V.

An equivalent formulation of normalcy is the following:

A topological space X is normal iff given a closed set C and an open set U containing C, there is an open set V containing C such that the closure of V is a subset of U.

Every metrizable space is normal, every compact Hausdorff space is normal, and every well ordered set is normal in the order topology on that set.

Normal spaces are in fact characterized by the Urysohn lemma, i.e., a space X is normal iff it satisfies the Urysohn lemma for every pair A, B of disjoint closed sets.

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