In topology and related branches of mathematics, normal spaces, T₄ spaces, and T₅ spaces are particularly nice kinds of topological spaces. These conditions are examples of separation axioms. Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In topology and related fields of mathematics, there are several restrictions that one often makes on the kinds of topological spaces that one wishes to consider. ... In topology and related branches of mathematics, the T0 spaces or Kolmogorov spaces form a broad class of well behaved topological spaces. ... The title given to this article is incorrect due to technical limitations. ... In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ... The definition used is this article is in contradiction with the usage of the term elsewhere in Wikipedia. ... In topology, completely Hausdorff spaces and Urysohn spaces are types of topological spaces satisfying slightly stronger separation axioms than the more familiar Hausdorff space. ... In topology and related fields of mathematics, regular spaces and T3 spaces are particularly nice kinds of topological spaces. ... In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are particularly nice kinds of topological spaces. ... In topology and related branches of mathematics, normal spaces, T4 spaces, and T5 spaces are particularly nice kinds of topological spaces. ... In topology and related branches of mathematics, normal spaces, T4 spaces, and T5 spaces are particularly nice kinds of topological spaces. ... In topology and related branches of mathematics, normal spaces, T4 spaces, and T5 spaces are particularly nice kinds of topological spaces. ... Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with spatial properties preserved under bicontinuous deformation (stretching without tearing or gluing); these are the topological invariants. ... Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In topology and related fields of mathematics, there are several restrictions that one often makes on the kinds of topological spaces that one wishes to consider. ...

Definitions

Suppose that X is a topological space. X is a normal space iff, 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. IFF, Iff or iff can stand for: Interchange File Format - a computer file format introduced by Electronic Arts Identification, friend or foe - a radio based identification system utilizing transponders iff - the mathematics concept if and only if International Flavors and Fragrances - a company producing flavors and fragrances International Freedom Foundation... In predicate logic, universal quantification is an attempt to formalise the notion that something (a logical predicate) is true for everything, or every relevant thing. ... In mathematics, two sets are said to be disjoint if they have no element in common. ... In topology and related branches of mathematics, a closed set is a set whose complement is open. ... In predicate logic, an existential quantification is the predication of a property or relation to at least one member of the domain. ... This is a glossary of some terms used in the branch of mathematics known as topology. ... In topology and related branches of mathematics, separated sets are pairs of subsets of a given topological space that are related to each other in a certain way. ...

Image File history File links Normal_space. ...

X is a T₄ space, if it's both normal and Hausdorff. In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ...

X is a completely normal space or a hereditarily 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. Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... It has been suggested that this article or section be merged with Logical biconditional. ... In topology and related branches of mathematics, separated sets are pairs of subsets of a given topological space that are related to each other in a certain way. ...

X is a T₅ space, or completely T₄ space, if it's both completely normal and Hausdorff, or equivalently, if every subspace of X is T₄.

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-delta set. Every perfectly normal space is automatically completely normal. In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ... In mathematics, the real line is simply the set of real numbers. ... In mathematics, the image of an element x in a set X under the function f : X → Y, denoted by f(x), is the unique y in Y that is associated with x. ... In mathematics, the unit interval is the interval [0,1], that is the set of all real numbers x such that zero is less than or equal to x and x is less than or equal to one. ... In the mathematical field of topology a G-delta set or Gδ set is a set in a topological space which is in a certain sense simple. ...

X is a perfectly T₄ space if it is both perfectly normal and Hausdorff.

Note that some mathematical literature uses different definitions for the terms "normal" and "T₄", 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 "T₅" always means the same as "completely T₄", whatever that may be.) For more on this issue, see History of the separation axioms. In general topology, the separation axioms have had a convoluted history, with many competing meanings for the same term, and many competing terms for the same concept. ...

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 T₄ 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 "T₄", or "completely normal Hausdorff" instead of "T₅". In topology and related fields of mathematics, regular spaces and T3 spaces are particularly nice kinds of topological spaces. ...

Fully normal spaces and fully T₄ spaces are discussed elsewhere; they are related to paracompactness. In mathematics, a paracompact space is a topological space in which every open cover admits an open locally finite refinement. ... In mathematics, a paracompact space is a topological space in which every open cover admits an open locally finite refinement. ... In mathematics, a paracompact space is a topological space in which every open cover admits an open locally finite refinement. ...

A locally normal space is a topological space where every point has an open neighbourhood that is normal. Every normal space is locally normal, but the converse is not true. A classical example of a completely regular locally normal space that is not normal is the Niemitzky plane.

Examples of normal spaces

Most spaces encountered in mathematical analysis are normal Hausdorff spaces, or at least normal regular spaces: Analysis is the generic name given to any branch of mathematics that depends upon the concepts of limits and convergence. ...

• All metric spaces (and hence all metrizable spaces) are perfectly normal Hausdorff;
• All pseudometric spaces (and hence all pseudometrisable spaces) are perfectly normal regular, although not in general Hausdorff;
• All compact Hausdorff spaces are normal;
• In particular, the Stone-Cech compactification of a Tychonoff space is normal Hausdorff;
• Generalizing the above examples, all paracompact Hausdorff spaces are normal, and all paracompact regular spaces are normal;
• In particular, all paracompact topological manifolds are normal Hausdorff (what about nonparacompact manifolds?);
• All order topologies on totally ordered sets are hereditarily normal and Hausdorff.
• Every regular second-countable space is completely normal, and every regular Lindelöf space is normal.

Also, all fully normal spaces are normal (even if not regular). Sierpinski space is an example of a normal space that isn't regular. In mathematics, a metric space is a set (or space) where a distance between points is defined. ... A metrizable space is a topological space that is homeomorphic to a metric space. ... In mathematics, a metric space is a set (or space) where a distance between points is defined. ... In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. ... In mathematics, compactification is applied to topological spaces to make them compact spaces. ... In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are particularly nice kinds of topological spaces. ... In mathematics, a paracompact space is a topological space in which every open cover admits an open locally finite refinement. ... In mathematics, a manifold M is a type of space, characterized in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. ... In mathematics, a total order or linear order on a set X is any binary relation on X that is antisymmetric, transitive, and total. ... In topology, a second-countable space is a topological space satisfying the second axiom of countability. Specifically, a space is said to be second-countable if its topology has a countable base. ... In mathematics, a Lindelöf space is a topological space in which every open cover has a countable subcover. ... In mathematics, a paracompact space is a topological space in which every open cover admits an open locally finite refinement. ... In topology, Sierpiński space S is the simplest example of a topological space that does not satisfy the T1 axiom. ...

Examples of non-normal spaces

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. This article needs to be cleaned up to conform to a higher standard of quality. ... In classical algebraic geometry (and to some extent also in modern algebraic geometry), the main objects of study are algebraic varieties. ... In abstract algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by Spec(R), is defined to be the set of all proper prime ideals of R. It is commonly augmented with the Zariski topology and with a structure sheaf, turning it into a locally ringed space. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with 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 A. H. Stone states that the product of uncountably many non-compact Hausdorff spaces is never normal. In mathematics a topological vector space is one of the basic structures investigated in functional analysis. ... Partial plot of a function f. ... In mathematics, the real line is simply the set of real numbers. ... Suppose { fn } is a sequence of functions sharing the same domain in common (for the moment, we defer making precise the nature of the values of these functions, but the reader may take them to be real numbers). ... In mathematics, an uncountable set is a set which is not countable. ... In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. ...

Properties

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: In topology and related areas of mathematics a continuous function is a morphism between topological spaces. ... In mathematics, the set of real numbers, denoted R, or in blackboard bold , is the set of all rational and irrational numbers. ... Partial plot of a function f. ...

The 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]. (In fancier terms, disjoint closed sets are not only separated by neighbourhoods, but also separated by a function.) Urysohns lemma in topology states that if X is a normal topological space and A and B are disjoint closed subsets of X, then there exists a continuous function from X into the unit interval [0, 1], f : X → [0, 1], such that f(a) = 0 for all... In mathematics, two sets are said to be disjoint if they have no element in common. ... In mathematics, the unit interval is the interval [0,1], that is the set of all real numbers x such that zero is less than or equal to x and x is less than or equal to one. ... In topology and related branches of mathematics, separated sets are pairs of subsets of a given topological space that are related to each other in a certain way. ...

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. The Tietze extension theorem in topology states that, if X is a normal topological space and f : A → R is a continuous map from a closed subset A of X into the real numbers carrying the standard topology, then there exists a continuous map F : X → R with...

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 mathematics, a cover of a set X is a collection of subsets C of X whose union is X. In symbols, if C = {Uα : α ∈ A} is an indexed family of subsets of X, then C is a cover if More generally, if Y is a subset... In mathematics, a partition of unity of a topological space X is a set of continuous functions {ρi} from X to the unit interval [0,1] such that every point has a neighbourhood where all but a finite number of the functions are identically zero, and the sum of all... In mathematics, a paracompact space is a topological space in which every open cover admits an open locally finite refinement. ...

In fact, any space that satisfies any one of these theorems must be normal.

A product of normal spaces is not necessarily normal. This fact was considered surprising when it was first proved by Robert Sorgenfrey. An example of this phenomenon is the Sorgenfrey plane. Also, a subset of a normal space need not be normal (i.e. not every normal Hausdorff space is a completely normal Hausdorff space), but examples of this are quite sophisticated to construct. In topology, the Sorgenfrey plane is a frequently-cited counterexample to many otherwise plausible-sounding conjectures. ...

Relationships to other separation axioms

If a normal space is R₀, then it is in fact completely regular. Thus, anything from "normal R₀" to "normal completely regular" is the same as what we normally call normal regular. Taking Kolmogorov quotients, we see that all normal T₁ spaces are Tychonoff. These are what we normally call normal Hausdorff spaces. In topology and related fields of mathematics, there are several restrictions that one often makes on the kinds of topological spaces that one wishes to consider. ... The title given to this article is incorrect due to technical limitations. ... In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are particularly nice kinds of topological spaces. ... In topology and related branches of mathematics, the T0 spaces or Kolmogorov spaces form a broad class of well behaved topological spaces. ... The title given to this article is incorrect due to technical limitations. ... In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are particularly nice kinds of topological 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. In topology, Sierpiński space S is the simplest example of a topological space that does not satisfy the T1 axiom. ...