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Suppose that X is a topological space.
X is a normal space if, given any disjoint closed sets E and F, there are a neighbourhood U of E and a neighbourhood V of F that are also disjoint. In fancier terms, this condition says that E and F can be separated by neighbourhoods.
X is a T_{4} space, if it's both normal and Hausdorff.
X is a completely normal space if every subspace of X is normal. It turns out that X is completely normal if and only if every two separated sets can be separated by neighbourhoods.
X is a T_{5} space, or completely T_{4} space, if it's both completely normal and Hausdorff, or equivalently, if every subspace of X is T_{4}.
X is a perfectly normal space if every two disjoint closed sets can be precisely separated by a function. That is, given disjoint closed sets E and F, there is a continuous function f from X to the real line R such the preimages of {0} and {1} under f are E and F respectively. You can also use the unit interval [0,1] in this definition; the result is the same. It turns out that X is perfectly normal if and only if X is normal and every closed set is a G_{δ} set. Every perfectly normal space is automatically completely normal.
X is a perfectly T_{4} space if it is both perfectly normal and Hausdorff.
Note that some mathematical literature uses different definitions for the terms "normal" and "T_{4}", and the terms containing those words. The definitions that we have given here are the ones usually used today, and the ones used in Wikipedia. However, some authors switch the meanings of the two terms in a given pair, or use both terms synonymously for only one condition, and you should take care to find out which definitions the author is using when reading mathematical literature. (But "T_{5}" always means the same as "completely T_{4}", whatever that may be.) For more on this issue, see History of the separation axioms[?].
You'll also find terms like normal regular space and normal Hausdorff space; these simply mean that the space both is normal and satisfies the other condition mentioned. In particular, a normal Hausdorff space is the same thing as a T_{4} space. These phrases are useful, since they're less ambiguous given the historical confusion of the terms' meanings. In Wikipedia, we prefer these phrases when applicable; that is, "normal Hausdorff" instead of "T_{4}", or "completely normal Hausdorff" instead of "T_{5}".
Fully normal spaces and fully T_{4} spaces are discussed elsewhere; they are related to paracompactness.
Most spaces encountered in mathematical analysis are normal Hausdorff spaces, or at least normal regular spaces:
Also, all fully normal spaces are normal (even if not regular). Sierpinski space[?] is an example of a normal space that isn't regular.
An important example of a nonnormal 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 nonnormal 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 A. H. Stone[?] states that the product of uncountably many noncompact Hausdorff spaces is never normal.
The main significance of normal spaces lies in the fact that they admit "enough" continuous realvalued functions, as expressed by the following theorems valid for any normal space X:
The Urysohn 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]. (In fancier terms, disjoint closed sets are not only separated by neighbourhoods, but also separated by a function.)
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 which extends f in the sense that F(x) = f(x) for all x in A.
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 theorems must be normal.
Relationships to other separation axioms
If a normal space is R_{0}, then it is in fact completely regular. Thus, anything from "normal R_{0}" to "normal completely regular" is the same as what we normally call normal regular. Taking Kolmogorov quotients, we see that all normal T_{1} spaces are Tychonoff. These are what we normally call normal Hausdorff spaces.
Counterexamples to some variations on these statements can be found in the lists above. Specifically, Sierpinski space is normal but not regular, while the space of functions from R to itself is Tychonoff but not normal.
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