In topology and related branches of mathematics, a normal space is a topological space X that satisfies Axiom T_{4}: every two disjoint closed sets of X have disjoint open neighborhoods. A normal Hausdorff space is also called a T_{4} space. These conditions are examples of separation axioms and their further strengthenings define completely normal Hausdorff spaces, or T_{5} spaces, and perfectly normal Hausdorff spaces, or T_{6} spaces.
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Definitions
A topological space X is a normal space if, given any disjoint closed sets E and F, there are open 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 T_{4} space is a T_{1} space X that is normal; this is equivalent to X being Hausdorff and normal.
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 T_{4} space, or T_{5} space is a completely normal Hausdorff topological space X; equivalently, every subspace of X must be a T_{4} space.
A perfectly normal space is a topological space X in which every two disjoint nonempty closed sets E and F can be 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 [0,1].)
It turns out that X is perfectly normal if and only if X is normal and every closed set is a G_{δ} 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.^{[citation needed]}
A Hausdorff perfectly normal space X is a T_{6} space, or perfectly T_{4} space.
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