## [Normality](/normality-0) Definition A separation axiom: a topological space X is normal if any two disjoint closed subsets of X have disjoint open neighborhoods. ## Principle Principle Closed sets can be separated by open neighborhoods; this strong separation enables extension and approximation theorems and provides control over continuous functions. ## Demonstration 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 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 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 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 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 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 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.