Normal space
Encyclopedia
In topology
and related branches of mathematics
, a normal space is a topological space
X that satisfies Axiom T4: every two disjoint closed set
s of X have disjoint open neighborhoods. A normal Hausdorff
space is also called a T4 space. These conditions are examples of separation axiom
s and their further strengthenings define completely normal Hausdorff spaces, or T5 spaces, and perfectly normal Hausdorff spaces, or T6 spaces.
X is a normal space if, given any disjoint closed set
s E and F, there are open neighbourhoods U of E and V of F that are also disjoint. More intuitively, this condition says that E and F can be separated by neighbourhoods.
A T4 space is a T1 space
X that is normal; this is equivalent to X being Hausdorff
and normal.
A completely normal space or a hereditarily normal space is a topological space X such that every subspace of X with subspace topology is a normal space. It turns out that X is completely normal if and only if every two separated sets can be separated by neighbourhoods.
A completely T4 space, or T5 space is a completely normal Hausdorff topological space X; equivalently, every subspace of X must be a T4 space.
A perfectly normal space is a topological space X in which every two disjoint non-empty closed sets E and F can be precisely separated by a continuous function
f from X to the real line
R: the preimages of {0} and {1} under f are, respectively, E and F. (In this definition, the real line can be replaced with the unit interval
[0,1].)
It turns out that X is perfectly normal if and only if X is normal and every closed set is a Gδ set
. Equivalently, X is perfectly normal if and only if every closed set is a zero set. Every perfectly normal space is automatically completely normal.
A Hausdorff perfectly normal space X is a T6 space, or perfectly T4 space.
Note that the terms "normal space" and "T4" and derived concepts occasionally have a different meaning. (Nonetheless, "T5" always means the same as "completely T4", whatever that may be.) The definitions given here are the ones usually used today, and the ones used in Wikipedia articles on separation axioms. For more on this issue, see History of the separation axioms
.
Terms like "normal regular space
" and "normal Hausdorff space" also turn up in the literature — they 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. Given the historical confusion of the meaning of the terms,
we prefer verbal descriptions when applicable, that is, "normal Hausdorff" instead of "T4", or "completely normal Hausdorff" instead of "T5".
Fully normal spaces and fully T4 spaces are discussed elsewhere; they are related to paracompactness.
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 Nemytskii plane.
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 is not regular.
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 function
s 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.
-valued function
s, 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].
(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 conditions must be normal.
A product of normal spaces is not necessarily normal. This fact 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), since every Tychonoff space is a subset of its Stone-Cech compactification (which is normal Hausdorff). A more explicit example is the Tychonoff plank
.
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 space
s 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.
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
and related branches of mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a normal space is a topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
X that satisfies Axiom T4: every two disjoint closed set
Closed set
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points...
s of X have disjoint open neighborhoods. A normal Hausdorff
Hausdorff space
In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...
space is also called a T4 space. These conditions are examples of separation axiom
Separation axiom
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. Some of these restrictions are given by the separation axioms...
s and their further strengthenings define completely normal Hausdorff spaces, or T5 spaces, and perfectly normal Hausdorff spaces, or T6 spaces.
Definitions
A topological spaceTopological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
X is a normal space if, given any disjoint closed set
Closed set
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points...
s E and F, there are open neighbourhoods U of E and V of F that are also disjoint. More intuitively, this condition says that E and F can be separated by neighbourhoods.
T1 space
In topology and related branches of mathematics, a T1 space is a topological space in which, for every pair of distinct points, each has an open neighborhood not containing the other. An R0 space is one in which this holds for every pair of topologically distinguishable points...
X that is normal; this is equivalent to X being Hausdorff
Hausdorff space
In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...
and normal.
A completely normal space or a hereditarily normal space is a topological space X such that every subspace of X with subspace topology is a normal space. It turns out that X is completely normal if and only if every two separated sets can be separated by neighbourhoods.
A completely T4 space, or T5 space is a completely normal Hausdorff topological space X; equivalently, every subspace of X must be a T4 space.
A perfectly normal space is a topological space X in which every two disjoint non-empty closed sets E and F can be precisely separated by a continuous function
Continuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
f from X to the real line
Real line
In mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the real line is the set of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one...
R: the preimages of {0} and {1} under f are, respectively, E and F. (In this definition, the real line can be replaced with the unit interval
Unit interval
In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1...
[0,1].)
It turns out that X is perfectly normal if and only if X is normal and every closed set is a Gδ set
G-delta set
In the mathematical field of topology, a Gδ set is a subset of a topological space that is a countable intersection of open sets. The notation originated in Germany with G for Gebiet meaning open set in this case and δ for Durchschnitt .The term inner limiting set is also used...
. Equivalently, X is perfectly normal if and only if every closed set is a zero set. Every perfectly normal space is automatically completely normal.
A Hausdorff perfectly normal space X is a T6 space, or perfectly T4 space.
Note that the terms "normal space" and "T4" and derived concepts occasionally have a different meaning. (Nonetheless, "T5" always means the same as "completely T4", whatever that may be.) The definitions given here are the ones usually used today, and the ones used in Wikipedia articles on separation axioms. For more on this issue, see History of the separation axioms
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.- Origins :...
.
Terms like "normal regular space
Regular space
In topology and related fields of mathematics, a topological space X is called a regular space if every non-empty closed subset C of X and a point p not contained in C admit non-overlapping open neighborhoods. Thus p and C...
" and "normal Hausdorff space" also turn up in the literature — they 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. Given the historical confusion of the meaning of the terms,
we prefer verbal descriptions when applicable, that is, "normal Hausdorff" instead of "T4", or "completely normal Hausdorff" instead of "T5".
Fully normal spaces and fully T4 spaces are discussed elsewhere; they are related to paracompactness.
A locally normal space
Locally normal space
In mathematics, particularly topology, a topological space X is locally normal if intuitively it looks locally like a normal space. More precisely, a locally normal space satisfies the property that each point of the space belongs to a neighbourhood of the space that is normal under the subspace...
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 Nemytskii plane.
Examples of normal spaces
Most spaces encountered in mathematical analysisMathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
are normal Hausdorff spaces, or at least normal regular spaces:
- All metric spaces (and hence all metrizable spaces) are perfectly normal Hausdorff;
- All pseudometric spacePseudometric spaceIn mathematics, a pseudometric space is a generalized metric space in which the distance between two distinct points can be zero. In the same way as every normed space is a metric space, every seminormed space is a pseudometric space...
s (and hence all pseudometrisable spaces) are perfectly normal regular, although not in general Hausdorff; - All compactCompact spaceIn mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
Hausdorff spaces are normal; - In particular, the Stone-Cech compactification of a Tychonoff spaceTychonoff spaceIn topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are kinds of topological spaces.These conditions are examples of separation axioms....
is normal Hausdorff; - Generalizing the above examples, all paracompact Hausdorff spaces are normal, and all paracompact regular spaces are normal;
- All paracompact topological manifoldTopological manifoldIn mathematics, a topological manifold is a topological space which looks locally like Euclidean space in a sense defined below...
s are perfectly normal Hausdorff. However, there exist non-paracompact manifolds which are not even normal. - All order topologies on totally ordered sets are hereditarily normal and Hausdorff.
- Every regular second-countable spaceSecond-countable spaceIn topology, a second-countable space, also called a completely separable space, is a topological space satisfying the second axiom of countability. A space is said to be second-countable if its topology has a countable base...
is completely normal, and every regular Lindelöf spaceLindelöf spaceIn mathematics, a Lindelöf space is a topological space in which every open cover has a countable subcover. The Lindelöf property is a weakening of the more commonly used notion of compactness, which requires the existence of a finite subcover....
is normal.
Also, all fully normal spaces are normal (even if not regular). Sierpinski space
Sierpinski space
In mathematics, the Sierpiński space is a finite topological space with two points, only one of which is closed.It is the smallest example of a topological space which is neither trivial nor discrete...
is an example of a normal space that is not regular.
Examples of non-normal spaces
An important example of a non-normal topology is given by the Zariski topologyZariski topology
In algebraic geometry, the Zariski topology is a particular topology chosen for algebraic varieties that reflects the algebraic nature of their definition. It is due to Oscar Zariski and took a place of particular importance in the field around 1950...
on an algebraic variety
Algebraic variety
In mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in algebraic geometry...
or on the spectrum of a ring
Spectrum of a ring
In abstract algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by Spec, is the set of all proper prime ideals of R...
, which is used in algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
.
A non-normal space of some relevance to analysis is the topological vector space
Topological vector space
In mathematics, a topological vector space is one of the basic structures investigated in functional analysis...
of all function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
s from the real line
Real line
In mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the real line is the set of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one...
R to itself, with the topology of pointwise convergence.
More generally, a theorem of A. H. Stone states that the product
Product topology
In topology and related areas of mathematics, a product space is the cartesian product of a family of topological spaces equipped with a natural topology called the product topology...
of uncountably many non-compact
Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
Hausdorff spaces is never normal.
Properties
The main significance of normal spaces lies in the fact that they admit "enough" continuous realReal number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
-valued function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
s, as expressed by the following theorems valid for any normal space X.
Urysohn's lemma
Urysohn's lemma
In topology, Urysohn's lemma is a lemma that states that a topological space is normal if and only if any two disjoint closed subsets can be separated by a function....
:
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
Unit interval
In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1...
[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 conditions must be normal.
A product of normal spaces is not necessarily normal. This fact was first proved by Robert Sorgenfrey
Robert Sorgenfrey
Robert Henry Sorgenfrey was an American mathematician and Professor Emeritus of Mathematics at the University of California, Los Angeles....
. 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), since every Tychonoff space is a subset of its Stone-Cech compactification (which is normal Hausdorff). A more explicit example is the Tychonoff plank
Tychonoff plank
In topology, the Tychonoff plank is a topological space that is a counterexample to several plausible-sounding conjectures. It is defined as the product of the two ordinal space[0,\omega_1]\times[0,\omega]...
.
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 space
T1 space
In topology and related branches of mathematics, a T1 space is a topological space in which, for every pair of distinct points, each has an open neighborhood not containing the other. An R0 space is one in which this holds for every pair of topologically distinguishable points...
s are Tychonoff
Tychonoff space
In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are kinds of topological spaces.These conditions are examples of separation axioms....
.
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
Sierpinski space
In mathematics, the Sierpiński space is a finite topological space with two points, only one of which is closed.It is the smallest example of a topological space which is neither trivial nor discrete...
is normal but not regular, while the space of functions from R to itself is Tychonoff but not normal.