Normal space

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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.

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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 T4 space is a T1 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 T4 space, or T5 space is a completely normal Hausdorff topological space X; equivalently, every subspace of X must be a T4 space.

A perfectly normal space is a topological space X in which every two disjoint non-empty 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 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.[citation needed]

A Hausdorff perfectly normal space X is a T6 space, or perfectly T4 space.

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