Normal space

In topology and related branches of mathematics, a normal space is a topological space X that satisfies Axiom T4: every two disjoint closed sets of X have disjoint open neighborhoods. A normal Hausdorff space is also called a T4 space. These conditions are examples of separation axioms and their further strengthenings define completely normal Hausdorff spaces, or T5 spaces, and perfectly normal Hausdorff spaces, or T6 spaces. A topological space X is a normal space if, given any disjoint closed sets E and F, there are neighbourhoods U of E and V of F that are also disjoint. More intuitively, this condition says that E and F can be separated by neighbourhoods. A T4 space is a T1 space X that is normal; this is equivalent to X being normal and Hausdorff. A completely normal space, or , is a topological space X such that every subspace of X with subspace topology is a normal space. It turns out that X is completely normal if and only if every two separated sets can be separated by neighbourhoods. Also, X is completely normal if and only if every open subset of X is normal with the subspace topology. A T5 space, or completely T4 space, is a completely normal T1 space X, which implies that X is Hausdorff; equivalently, every subspace of X must be a T4 space. A perfectly normal space is a topological space in which every two disjoint closed sets and can be precisely separated by a function, in the sense that there is a continuous function from to the interval such that and . This is a stronger separation property than normality, as by Urysohn's lemma disjoint closed sets in a normal space can be separated by a function, in the sense of and , but not precisely separated in general. It turns out that X is perfectly normal if and only if X is normal and every closed set is a Gδ set. Equivalently, X is perfectly normal if and only if every closed set is the zero set of a continuous function. The equivalence between these three characterizations is called Vedenissoff's theorem.