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. 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 a hereditarily normal space 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. A completely T4 space, or T5 space is a completely normal T1 space topological 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 X in which every two disjoint closed sets E and F can be precisely separated by a continuous function f from X to the real line R: the preimages of {0} and {1} under f are, respectively, E and F. (In this definition, the real line can be replaced with the unit interval .) 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 a zero set. Every perfectly normal space is automatically completely normal. A Hausdorff perfectly normal space X is a T6 space, or perfectly T4 space. Note that the terms 'normal space' and 'T4' and derived concepts occasionally have a different meaning. (Nonetheless, 'T5' always means the same as 'completely T4', whatever that may be.) The definitions given here are the ones usually used today. For more on this issue, see History of the separation axioms.