Normal space
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.
Suppose that
X is a topological space.
X is a
normal space ,
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.
 |
The closed sets E and F, here represented by closed disks on opposite sides of the picture, are separated by their respective neighbourhoods U and V, here represented by larger, but still disjoint, open disks. |
X is a
T4 space, if it's both normal and
Hausdorff.
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.
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 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-delta set.Every perfectly normal space is automatically completely normal.
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 "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 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
Niemitzky plane.
Most spaces encountered in
mathematical analysis are normal Hausdorff spaces, or at least normal regular spaces:
* 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.
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
A. H. Stone states that the
product of
uncountably many non-
compact Hausdorff spaces is never normal.
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:
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.)
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.
A
product of normal spaces is not necessarily normal. This fact was considered surprising when it was first proved by
Robert Sorgenfrey.
If a normal space is
R0, 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
T1 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.
