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.
[1] This is a stronger separation property than normality, as by Urysohn's lemma disjoint closed sets in a normal space can be separated by a function, in the sense of
The equivalence between these three characterizations is called Vedenissoff's theorem.
Note that the terms "normal space" and "T4" and derived concepts occasionally have a different meaning.
Given the historical confusion of the meaning of the terms, verbal descriptions when applicable are helpful, that is, "normal Hausdorff" instead of "T4", or "completely normal Hausdorff" instead of "T5".
An important example of a non-normal topology is given by the Zariski topology on an algebraic variety or on the spectrum of a ring, which is used in algebraic geometry.
A non-normal space of some relevance to analysis is the topological vector space of all functions from the real line R to itself, with the topology of pointwise convergence.
More generally, a theorem of Arthur Harold Stone states that the product of uncountably many non-compact metric spaces is never normal.
[6] The main significance of normal spaces lies in the fact that they admit "enough" continuous real-valued functions, as expressed by the following theorems valid for any normal space X. Urysohn's lemma: If A and B are two disjoint closed subsets of X, then there exists a continuous function f from X to the real line R such that f(x) = 0 for all x in A and f(x) = 1 for all x in B.
In fact, we can take the values of f to be entirely within the unit interval [0,1].
More generally, the Tietze extension theorem: If A is a closed subset of X and f is a continuous function from A to R, then there exists a continuous function F: X → R that extends f in the sense that F(x) = f(x) for all x in A.
has the lifting property with respect to a map from a certain finite topological space with five points (two open and three closed) to the space with one open and two closed points.
[7] If U is a locally finite open cover of a normal space X, then there is a partition of unity precisely subordinate to U.
This shows the relationship of normal spaces to paracompactness.
In fact, any space that satisfies any one of these three conditions must be normal.
The only large class of product spaces of normal spaces known to be normal are the products of compact Hausdorff spaces, since both compactness (Tychonoff's theorem) and the T2 axiom are preserved under arbitrary products.
[8] If a normal space is R0, then it is in fact completely regular.
Taking Kolmogorov quotients, we see that all normal T1 spaces are Tychonoff.
Every normal space is pseudonormal, but not vice versa.
