A separation axiom: a topological space X is normal if any two disjoint closed subsets of X have disjoint open neighborhoods.

Principle

Closed sets can be separated by open neighborhoods; this strong separation enables extension and approximation theorems and provides control over continuous functions.

Demonstration

Every metric space is normal: given two disjoint closed sets in R, one constructs disjoint open intervals or uses distance functions to build separating opens and continuous functions.

Misapplication

Assuming normality is hereditary to arbitrary subspaces or preserved by arbitrary products is a misuse: subspaces need not be normal and infinite products of normal spaces can fail to be normal.

Consequence

Under the usual convention that normal includes T1 (sometimes denoted T4), normality yields Urysohn-type separation and the Tietze extension theorem for continuous real-valued functions on closed sets.

Reversal

A non-normal space contains two disjoint closed sets that cannot be separated by disjoint opens; such failure obstructs extension of continuous functions and fine separation arguments.

Boundary

Normality is a global separation condition for closed sets in topological spaces. Conventions vary as to whether T1 is included in the definition; it is distinct from weaker axioms like regularity or complete regularity.

Semantic Tension

Normality is often conflated with regularity or complete regularity; the tension lies in whether separation is required for points and closed sets (regular) or for closed sets only (normal) and whether T1/Hausdorff assumptions are present.

Synthesis

Normality asserts that disjoint closed sets admit disjoint open neighborhoods, giving a global level of separability that supports extension theorems and construction of continuous partitions of unity under appropriate hypotheses.