In the mathematical discipline of
general topology
,
Stone??ech compactification
(or
?ech?Stone compactification
[1]
) is a technique for constructing a
universal map
from a
topological space
X
to a
compact
Hausdorff space
βX
. The Stone??ech compactification
βX
of a topological space
X
is the largest, most general compact Hausdorff space "generated" by
X
, in the sense that any continuous map from
X
to a compact Hausdorff space
factors through
βX
(in a unique way). If
X
is a
Tychonoff space
then the map from
X
to its
image
in
βX
is a
homeomorphism
, so
X
can be thought of as a (
dense
) subspace of
βX
; every other compact Hausdorff space that densely contains
X
is a
quotient
of
βX
. For general topological spaces
X
, the map from
X
to
βX
need not be
injective
.
A form of the
axiom of choice
is required to prove that every topological space has a Stone??ech compactification. Even for quite simple spaces
X
, an accessible concrete description of
βX
often remains elusive. In particular, proofs that
βX
\
X
is nonempty do not give an explicit description of any particular point in
βX
\
X
.
The Stone??ech compactification occurs implicitly in a paper by
Andrey Nikolayevich Tychonoff
(
1930
) and was given explicitly by
Marshall Stone
(
1937
) and
Eduard ?ech
(
1937
).
History
[
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]
Andrey Nikolayevich Tikhonov
introduced completely regular spaces in 1930 in order to avoid the pathological situation of
Hausdorff spaces
whose only continuous
real
-valued functions are constant maps.
In the same 1930 article where Tychonoff defined completely regular spaces, he also proved that every
Tychonoff space
(i.e.
Hausdorff
completely regular space) has a Hausdorff
compactification
(in this same article, he also proved
Tychonoff's theorem
).
In 1937, ?ech extended Tychonoff's technique and introduced the notation β
X
for this compactification.
Stone also constructed β
X
in a 1937 article, although using a very different method.
Despite Tychonoff's article being the first work on the subject of the Stone??ech compactification and despite Tychonoff's article being referenced by both Stone and ?ech, Tychonoff's name is rarely associated with β
X
.
Universal property and functoriality
[
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]
The Stone??ech compactification of the topological space
X
is a compact Hausdorff space
βX
together with a continuous map
i
X
:
X
→
βX
that has the following
universal property
: any
continuous map
f
:
X
→
K
, where
K
is a compact Hausdorff space, extends uniquely to a continuous map
βf
:
βX
→
K
, i.e. (
βf
)
i
X
=
f
.
The universal property of the Stone-Cech compactification expressed in diagram form.
As is usual for universal properties, this universal property characterizes
βX
up to
homeomorphism
.
As is outlined in
§ Constructions
, below, one can prove (using the axiom of choice) that such a Stone??ech compactification
i
X
:
X
→
βX
exists for every topological space
X
. Furthermore, the image
i
X
(
X
) is dense in
βX
.
Some authors add the assumption that the starting space
X
be Tychonoff (or even
locally compact
Hausdorff), for the following reasons:
- The map from
X
to its image in
βX
is a homeomorphism if and only if
X
is Tychonoff.
- The map from
X
to its image in
βX
is a homeomorphism to an open subspace if and only if
X
is locally compact Hausdorff.
The Stone??ech construction can be performed for more general spaces
X
, but in that case the map
X
→
βX
need not be a homeomorphism to the image of
X
(and sometimes is not even injective).
As is usual for universal constructions like this, the extension property makes
β
a
functor
from
Top
(the
category of topological spaces
) to
CHaus
(the category of compact Hausdorff spaces). Further, if we let
U
be the
inclusion functor
from
CHaus
into
Top
, maps from
βX
to
K
(for
K
in
CHaus
) correspond
bijectively
to maps from
X
to
UK
(by considering their
restriction
to
X
and using the universal property of
βX
). i.e.
- Hom(
βX
,
K
) ? Hom(
X
,
UK
),
which means that
β
is
left adjoint
to
U
. This implies that
CHaus
is a
reflective subcategory
of
Top
with reflector
β
.
Examples
[
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]
If
X
is a compact Hausdorff space, then it coincides with its Stone??ech compactification.
The Stone??ech compactification of the
first uncountable ordinal
, with the
order topology
, is the ordinal
. The Stone??ech compactification of the
deleted Tychonoff plank
is the Tychonoff plank.
[6]
Constructions
[
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]
Construction using products
[
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]
One attempt to construct the Stone??ech compactification of
X
is to take the closure of the image of
X
in
![{\displaystyle \prod \nolimits _{f:X\to K}K}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6081ef61d79b6cc1ee12eb255e9d49e6e358c8a3)
where the product is over all maps from
X
to compact Hausdorff spaces
K
(or, equivalently, the image of
X
by the right
Kan extension
of the identity functor of the category
CHaus
of compact Hausdorff spaces along the inclusion functor of
CHaus
into the category
Top
of general topological spaces). By
Tychonoff's theorem
this product of compact spaces is compact, and the closure of
X
in this space is therefore also compact. This works intuitively but fails for the technical reason that the collection of all such maps is a
proper class
rather than a set. There are several ways to modify this idea to make it work; for example, one can restrict the compact Hausdorff spaces
K
to have underlying set
P
(
P
(
X
)) (the
power set
of the power set of
X
), which is sufficiently large that it has cardinality at least equal to that of every compact Hausdorff space to which
X
can be mapped with dense image.
Construction using the unit interval
[
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]
One way of constructing
βX
is to let
C
be the set of all
continuous functions
from
X
into [0, 1] and consider the map
where
![{\displaystyle e(x):f\mapsto f(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/560279d9664903e3471e124ec6bf95c6a0289094)
This may be seen to be a continuous map onto its image, if [0, 1]
C
is given the
product topology
. By
Tychonoff's theorem
we have that [0, 1]
C
is compact since [0, 1] is. Consequently, the closure of
X
in [0, 1]
C
is a compactification of
X
.
In fact, this closure is the Stone??ech compactification. To verify this, we just need to verify that the closure satisfies the appropriate universal property. We do this first for
K
= [0, 1], where the desired extension of
f
:
X
→ [0, 1] is just the projection onto the
f
coordinate in [0, 1]
C
. In order to then get this for general compact Hausdorff
K
we use the above to note that
K
can be embedded in some cube, extend each of the coordinate functions and then take the product of these extensions.
The special property of the
unit interval
needed for this construction to work is that it is a
cogenerator
of the category of compact Hausdorff spaces: this means that if
A
and
B
are compact Hausdorff spaces, and
f
and
g
are distinct maps from
A
to
B
, then there is a map
h
:
B
→ [0, 1] such that
hf
and
hg
are distinct. Any other cogenerator (or cogenerating set) can be used in this construction.
Construction using ultrafilters
[
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]
Alternatively, if
is
discrete
, then it is possible to construct
as the set of all
ultrafilters
on
with the elements of
corresponding to the
principal ultrafilters
. The topology on the set of ultrafilters, known as the
Stone topology
, is generated by sets of the form
for
a subset of
Again we verify the universal property: For
with
compact Hausdorff and
an ultrafilter on
we have an
ultrafilter base
on
the
pushforward
of
This has a unique
limit
because
is compact Hausdorff, say
and we define
This may be verified to be a continuous extension of
Equivalently, one can take the
Stone space
of the
complete Boolean algebra
of all subsets of
as the Stone??ech compactification. This is really the same construction, as the Stone space of this Boolean algebra is the set of ultrafilters (or equivalently prime
ideals
, or homomorphisms to the 2 element Boolean algebra) of the Boolean algebra, which is the same as the set of ultrafilters on
The construction can be generalized to arbitrary Tychonoff spaces by using
maximal filters
of
zero sets
instead of ultrafilters.
[7]
(Filters of closed sets suffice if the space is
normal
.)
Construction using C*-algebras
[
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]
The Stone??ech compactification is naturally homeomorphic to the
spectrum
of C
b
(
X
).
[8]
Here C
b
(
X
) denotes the
C*-algebra
of all continuous bounded
complex-valued functions
on
X
with
sup-norm
. Notice that C
b
(
X
) is canonically isomorphic to the
multiplier algebra
of C
0
(
X
).
The Stone??ech compactification of the natural numbers
[
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]
In the case where
X
is
locally compact
, e.g.
N
or
R
, the image of
X
forms an open subset of
βX
, or indeed of any compactification, (this is also a necessary condition, as an open subset of a compact Hausdorff space is locally compact). In this case one often studies the
remainder
of the space,
βX
\
X
. This is a closed subset of
βX
, and so is compact. We consider
N
with its
discrete topology
and write
β
N
\
N
=
N
* (but this does not appear to be standard notation for general
X
).
As explained above, one can view
β
N
as the set of
ultrafilters
on
N
, with the topology generated by sets of the form
for
U
a subset of
N
. The set
N
corresponds to the set of
principal ultrafilters
, and the set
N
* to the set of
free ultrafilters
.
The study of
β
N
, and in particular
N
*, is a major area of modern
set-theoretic topology
. The major results motivating this are
Parovicenko's theorems
, essentially characterising its behaviour under the assumption of the
continuum hypothesis
.
These state:
- Every compact Hausdorff space of
weight
at most
(see
Aleph number
) is the continuous image of
N
* (this does not need the continuum hypothesis, but is less interesting in its absence).
- If the continuum hypothesis holds then
N
* is the unique
Parovicenko space
, up to isomorphism.
These were originally proved by considering
Boolean algebras
and applying
Stone duality
.
Jan van Mill has described
β
N
as a "three headed monster"?the three heads being a smiling and friendly head (the behaviour under the assumption of the continuum hypothesis), the ugly head of independence which constantly tries to confuse you (determining what behaviour is possible in different models of set theory), and the third head is the smallest of all (what you can prove about it in
ZFC
).
[9]
It has relatively recently been observed that this characterisation isn't quite right?there is in fact a fourth head of
β
N
, in which
forcing axioms
and Ramsey type axioms give properties of
β
N
almost diametrically opposed to those under the continuum hypothesis, giving very few maps from
N
* indeed. Examples of these axioms include the combination of
Martin's axiom
and the
Open colouring axiom
which, for example, prove that (
N
*)
2
≠
N
*, while the continuum hypothesis implies the opposite.
An application: the dual space of the space of bounded sequences of reals
[
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]
The Stone??ech compactification
β
N
can be used to characterize
(the
Banach space
of all bounded sequences in the scalar
field
R
or
C
, with
supremum norm
) and its
dual space
.
Given a bounded sequence
there exists a
closed ball
B
in the scalar field that contains the image of
.
is then a function from
N
to
B
. Since
N
is discrete and
B
is compact and Hausdorff,
a
is continuous. According to the universal property, there exists a unique extension
βa
:
β
N
→
B
. This extension does not depend on the ball
B
we consider.
We have defined an extension map from the space of bounded scalar valued sequences to the space of continuous functions over
β
N
.
![{\displaystyle \ell ^{\infty }(\mathbf {N} )\to C(\beta \mathbf {N} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a9e73ba67aec82724f8626266feade4b29cd94e6)
This map is bijective since every function in
C
(
β
N
) must be bounded and can then be restricted to a bounded scalar sequence.
If we further consider both spaces with the sup norm the extension map becomes an
isometry
. Indeed, if in the construction above we take the smallest possible ball
B
, we see that the sup norm of the extended sequence does not grow (although the image of the extended function can be bigger).
Thus,
can be identified with
C
(
β
N
). This allows us to use the
Riesz representation theorem
and find that the dual space of
can be identified with the space of finite
Borel measures
on
β
N
.
Finally, it should be noticed that this technique generalizes to the
L
∞
space of an arbitrary
measure space
X
. However, instead of simply considering the space
βX
of ultrafilters on
X
, the right way to generalize this construction is to consider the
Stone space
Y
of the measure algebra of
X
: the spaces
C
(
Y
) and
L
∞
(
X
) are isomorphic as C*-algebras as long as
X
satisfies a reasonable finiteness condition (that any set of positive measure contains a subset of finite positive measure).
A monoid operation on the Stone??ech compactification of the naturals
[
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]
The
natural numbers
form a
monoid
under
addition
. It turns out that this operation can be extended (generally in more than one way, but uniquely under a further condition) to
β
N
, turning this space also into a monoid, though rather surprisingly a non-commutative one.
For any subset,
A
, of
N
and a positive integer
n
in
N
, we define
![{\displaystyle A-n=\{k\in \mathbf {N} \mid k+n\in A\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8fd6157c07da63a087f870a29ac95588bb945e6d)
Given two ultrafilters
F
and
G
on
N
, we define their sum by
![{\displaystyle F+G={\Big \{}A\subseteq \mathbf {N} \mid \{n\in \mathbf {N} \mid A-n\in F\}\in G{\Big \}};}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9a07a0d1bd7c413868d6dd6a45e350a7fe3b6e23)
it can be checked that this is again an ultrafilter, and that the operation + is
associative
(but not commutative) on β
N
and extends the addition on
N
; 0 serves as a neutral element for the operation + on
β
N
. The operation is also right-continuous, in the sense that for every ultrafilter
F
, the map
![{\displaystyle {\begin{cases}\beta \mathbf {N} \to \beta \mathbf {N} \\G\mapsto F+G\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/081435f305075526152447d1f49498399ca1f81b)
is continuous.
More generally, if
S
is a
semigroup
with the discrete topology, the operation of
S
can be extended to
βS
, getting a right-continuous associative operation.
[10]
See also
[
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]
Notes
[
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]
- ^
M. Henriksen, "Rings of continuous functions in the 1950s", in
Handbook of the History of General Topology
, edited by C. E. Aull, R. Lowen,
Springer Science & Business Media, 2013, p. 246
- ^
Walker, R. C. (1974).
The Stone-?ech Compactification
. Springer. pp. 95?97.
ISBN
978-3-642-61935-9
.
- ^
W.W. Comfort, S. Negrepontis,
The Theory of Ultrafilters
, Springer, 1974.
- ^
This is Stone's original construction.
- ^
van Mill, Jan (1984), "An introduction to βω", in Kunen, Kenneth; Vaughan, Jerry E. (eds.),
Handbook of Set-Theoretic Topology
, North-Holland, pp. 503?560,
ISBN
978-0-444-86580-9
- ^
Hindman, Neil; Strauss, Dona (2011-01-21).
Algebra in the Stone-Cech Compactification
. Berlin, Boston: DE GRUYTER.
doi
:
10.1515/9783110258356
.
ISBN
978-3-11-025835-6
.
References
[
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]
- ?ech, Eduard
(1937), "On bicompact spaces",
Annals of Mathematics
,
38
(4): 823?844,
doi
:
10.2307/1968839
,
hdl
:
10338.dmlcz/100420
,
JSTOR
1968839
- Dunford, Nelson
;
Schwarz, Jacob T.
(1988).
Linear Operators part I:general theory
(Wiley Classics ed.). John Wiley & Sons. p. 276.
- Hindman, Neil
;
Strauss, Dona
(1998),
Algebra in the Stone?Cech compactification. Theory and applications
, de Gruyter Expositions in Mathematics, vol. 27 (2nd revised and extended 2012 ed.), Berlin: Walter de Gruyter & Co., pp. xiv+485 pp,
doi
:
10.1515/9783110809220
,
ISBN
978-3-11-015420-7
,
MR
1642231
- Munkres, James R.
(2000).
Topology
(Second ed.).
Upper Saddle River, NJ
:
Prentice Hall, Inc
.
ISBN
978-0-13-181629-9
.
OCLC
42683260
.
- Koshevnikova, I.G. (2001) [1994],
"Stone-?ech compactification"
,
Encyclopedia of Mathematics
,
EMS Press
- Narici, Lawrence; Beckenstein, Edward (2011).
Topological Vector Spaces
. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press.
ISBN
978-1584888666
.
OCLC
144216834
.
- Shields, Allen (1987), "Years ago",
Mathematical Intelligencer
,
9
(2): 61?63,
doi
:
10.1007/BF03025901
,
S2CID
189886579
- Stone, Marshall H. (1937), "Applications of the theory of Boolean rings to general topology",
Transactions of the American Mathematical Society
,
41
(3): 375?481,
doi
:
10.2307/1989788
,
JSTOR
1989788
- Tychonoff, Andrey
(1930), "Uber die topologische Erweiterung von Raumen",
Mathematische Annalen
,
102
: 544?561,
doi
:
10.1007/BF01782364
,
ISSN
0025-5831
,
S2CID
124737286
External links
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]