1. Introduction
The foundation stone for Nonstandard analysis was laid by Abraham Robinson in 1966 with the publication of his book “Nonstandard analysis” [1]. Here R is extended to an ordered field extension
of R, where “very small” (infinitesimals [2] [3] ) elements and “very large” (infinite) elements exist. Elements of
are called hyper-real numbers. Sets, relations, functions in R get extended to
in a natural way. Also statements in R get extended to
, with suitable interpretation, with the aid of a pivotal principle called the Transfer principle. On the other hand, certain statements in
hold good in R by the Downward Transfer principle. The whole idea is not so much to learn the extension as is to study more of the original frame R by getting on a higher platform
. Moreover, the higher platform renders a natural way of looking at the concepts in R and hence provides a natural and shorter proof of even the already proved statements in R. Nonstandard analysis propounded by Robinson is not just a formalization of the infinitesimals in R. The type of extension from R to
applies to any mathematical structure based on a suitable nonstandard model. For instance, if X is a topological space or a linear space or a measure space, it gets extended to
allowing extensions of sets, relations, functions etc., and providing Transfer and Downward Transfer theorems to carry out all algebra and analysis [4]. Again, as in R, the idea is to learn more about X and not so much about
.
The purpose of this article is to present nonstandard versions of three classical compactifications: Stone-Cech compactification of a completely regular space, Alexandroff’s one-point compactification of a locally compact Hausdorff space and Bohr compactification of a locally compact Abelian group. For the classical versions of Stone-Cech compactification and Alexandroff’s one-point compactification and for other preliminaries in topology, we refer to [5] [6] [7]. For Bohr compactification and the preliminaries on locally compact Abelian groups, we refer to [8]. The procedure for nonstandard Stone-Cech compactification is followed for uniform spaces in [9]. This is in analogy with the nonstandard completion of a uniform space mimicking the nonstandard completion procedure of a metric space [4] [10]. Since the basic principle involved is the same in all these, it is contextual to present these in a single article. The general procedure is as indicated in the abstract. These compactifications come under the general category of Q-compactifications [4] for suitable families of functions Q. However, details have to be worked out individually and this has been done systematically in this article.
2. Main Results
Definition 2.1.
Let X be a topological space. A topological space
is said to be a compactification of X if X can be embedded in
as a dense subspace. That is, if there exists a homeomorphism
such that range
is dense in
.
First we take up the nonstandard version of the Stone-Cech compactification of a completely regular space. We have the following.
Theorem 2.2.
Let X be a completely regular space. Then it has a compactification
.
Proof.
Let
be the set of bounded continuous real valued functions on X.
In
define
if
Clearly ~ is an equivalence relation on
.
We denote the equivalence class of
by
and the set of equivalence classes by
.
we define
by
.
We consider
with the weak topology
generated by the
.
We wish to show that
defined by
is an embedding of X as a dense subspace of
.
(i) First we see that
is one-one.
Let
for
Then
Therefore
That is,
Therefore
, by complete regularity of X
Therefore
is one-one.
(ii) To prove:
is continuous
Let
A basic neighbourhood of
is given by
is open in X, by the continuity of each
.
Hence
is continuous.
(iii) To prove :
is an open map
Let V be a basic neighbourhood of
, where
By complete regularity of X, fix
such that
,
and
Now
(1)
Let
with
,
Then
Therefore
, by (1).
Therefore
(2)
Also
is a neighbourhood of
in
This neighbourhood
,
since
.
Also this neighbourhood
, by (2)
Therefore
is open in
Hence
is an open map.
Thus
is a homeomorphism of X onto
(iv) To prove:
is dense in
.
Let
We take a basic neighbourhood of
given by
Now
, since it contains y.
Therefore
, by Downward Transfer.
That is,
Therefore
for some
Therefore
is dense in
(v) To prove:
is compact.
For each
, we associate a map
from
to R defined by
Let A be the range of T.
Claim: T is a one-one mapping of
onto A.
and
are not infinitely close to each other for some
Therefore
Therefore T is one-one, establishing the claim.
Define a topology on A by declaring U open in A if
is open in
.
By definition, T is a homeomorphism of
onto A.
To show
is compact, all we need to show is that A is compact.
A basic neighbourhood of
is of the form
,
where
and
Since X is dense in
,
such that
That is,
for
That is,
for
That is,
in R,
such that
for
By concurrence,
such that
in
,
Taking
as a positive infinitesimal, we get the following:
such that
,
(3)
Now let
To show A is compact, we need to show that
is near some
By (3),
such that
,
Take
Then
,
Therefore
Therefore A is compact.
This completes the proof.
□
Next in line is Alexandroff’s one-point compactification of a locally compact Hausdorff space.
Let
be the set of continuous real valued functions on X with compact support.
Theorem 2.3.
Let X be a non-compact locally compact Hausdorff space. By replacing
by
in the proof of last theorem, we get
with the weak topology generated by the
,
. Then
is a singleton set and
turns out to be the Alexandroff one-point compactification of X.
Proof.
In
define
if
Clearly ~ is an equivalence relation on
.
We denote the equivalence class of
by
and the set of equivalence classes by
. X is imbedded in
by the mapping
for
.
we define
by
.
We consider
with the weak topology
generated by the
.
(i) First we show that
, take
such that
for
By local compactness,
such that
and
for
Thus
such that y is not related to
under ~ for
By Concurrence,
such that
is not related to x under ~
Therefore
and hence
(ii) Next we show that
is singleton
Let
and
with
, a compact set in X.
Then
; otherwise
for some
and
, contradicting
Therefore
Thus if
, then
Hence
Therefore
Therefore
is a singleton set, say
Thus
, where X is with the identification
From classical analysis, we have
is compact with the one-point compactification. Let this topology be denoted by
.
Now we wish to show
(iii) First we show that every set open in the locally compact Hausdorff space X is open in
with the
-topology.
For this it is enough to show X is open in
Take any
an open neighbourhood U of x such that
is compact, by local compactness of X.
Again we can find an open neighbourhood V of x such that
.
such that
.
In particular,
.
Since
Then
is a
-open neighbourhood of
contained in X.
Therefore X is open in
and hence any U open in X is open in
as well.
(iv) Take any set K compact in X.
We show
is open in
Take any
We need to show that
is an interior point of
Case (a):
By local compactness, there exists an open set U inX such that
and
is compact.
such that
Since
Then
is a
-open neighbourhood of
contained in
.
Case (b):
Then
for some
Using local compactness as before,
such that
Then
is a
-open neighbourhood of
contained
in
Therefore in any case
is open in
From (iii) and (iv) we get
(v) For the other way implication, take any basic open set
, where G is open in R and
Case (a): Suppose
Then
is
with the identification
and so is open X and hence in
with the
-topology.
Case (b): Suppose
Then
, where K is the compact set
Hence
is open in
with the
-topology.
Therefore
and hence
Thus the weak topology generated by the
, where
, is the Alexandroff one-point compactification of X. □
Finally we take up Bohr Compactification of a locally compact abelian group.
Let G be a non-compact locally compact abelian group with dual group
. On
define a relation ~ by
if
. ~ is an equivalence relation on
. Denote the equivalence class of
by
and the set of equivalence classes by
. Define addition in
by
. Define
on
by
. Then
is well-defined on
. Topologize
by the weak topology generated by the
.
Theorem 2.4.
The above mentioned
is a compact abelian group and is a compactification of the locally compact abelian group G.
Proof.
For most part, the proof mimicks that of Theorem 2.2. Since each
is finite (in fact
),
is defined.
If
, then
and hence
so that
is well-defined.
Clearly
is an abelian group.
Next we wish to show that + and − are continuous on
.
Let
and
.
Therefore
Therefore + is continuous on
Similarly − is continuous.
Claim 1: The map
is one to one.
for some
, since
separates points of G.
Therefore we have claim 1.
Claim 2: The map
is continuous.
Therefore
in
Hence the map
is continuous.
Claim 3: The map
imbeds G as a dense subgroup of
Let
We take a basic neighbourhood of
given by
Now
, since it contains y.
Therefore
, by Downward Transfer.
That is,
Therefore there exists
for some
This establishes claim 3.
Claim 4:
is compact.
For each
, associate a map
from
to C defined by
Let A be the range of T.
First we see that T is a one-one mapping of
onto A.
is not infinitely close to
for some
T is one-one.
Now we define a topology on A by declaring U open in A if
is open in
.
By definition, T is a homeomorphism of
onto A.
To show
is compact, it is enough to show that A is compact.
A basic neighbourhood of
is of the form
where
and
Since G is dense in
,
such that
That is,
for
That is,
for
That is,
,
,
in R,
such that
for
By concurrence,
such that
in
,
Taking
as a positive infinitesimal we get the following:
such that
-------(*)
Now let
.
To show A is compact, what we need to show is that
is near some
By (*),
such that
,
Take
Then
,
Therefore
Therefore A is compact.
This completes the proof. □