Download 1980 Mathematical Subject Classification Code. 54A20

I nternat. J. Math. ath. Sci.
Vol. 2 #3 (1979) 345-368
345
COMPACTIFICATION$
OF CONVERGENCE SPACES
D. C. KENT
Department of Pure and Applied Mathematics
Washington State University
Pullman, Washington 99163
G. D. RICHARDSON
Department of Mathematics
East Carolina University
Greenville, North Carolina 27834
(Received May 15, 1979)
ABSTRACT.
This paper summarizes most of the results to date on convergence space
compactifications, and establishes necessary and sufficient conditions for the
existence of largest and smallest compactifications subject to various conditions
imposed upon the compactifications.
KEY WORDS AND PHRASES. Convergence space, compactfication, locally bounded space,
setially compact space, relatively diagonal compatification, simple ompacti-
fication.
1980 Mathematical Subject
i.
Classification
Code.
54A20, 54D5.
INTRODUCTION.
The subject of convergence space compactifications is now about ten years old,
although some related concepts, such as Novak’s sequential envelope [13], are of
D.C. KENT and G. D. RICHARDSON
346
earlier vintage.
Our goal is to summarize the results in this area which have been
In the latter
obtained to date, and to give further development to the subject.
endeavor, we follow a path initiated by C. J. M. Rao, making use of ideas introduced
by Ellen Reed and inspired by the completion theory of H. Kowalsky.
Convergence spaces were originally defined in terms of sequences by M. Frechet
[5]
in 1906.
A compactlflcatlon theory for convergence spaces had to await the
The foundational
development of convergence spaces defined by means of filters.
papers for filter convergence spaces were written by G. Choquet [i] in 1948,
H. Kowalsky [12] in 1954, and H. Fischer [4] in 1959.
In 1970, G. D. Richardson [21] and J. F. Ramaley and O. Wyler [15] published
different versions of a
"Stone-ech compactlflcatlon"
the former paper, each
T2
for convergence spaces.
convergence space is embeded in a compact
gence space with the property that each map into a compact
to the compactlflcatlon space.
2
conver-
space can be lifted
3
Similar results are obtained in the latter paper,
but with the following significant differences:
T 3,
T
T
In
the "compactlflcatlon" space is
but the injection map into this space is not an embedding.
Thus the Ramaley-
Wyler "compactlficatlon" is not a compactlflcation in the sense that we use the term
The universal property for Richardson’s compactlflcation is not entirely
T 3.
satisfactory because the compactiflcation space is usually not
R. Gazlk [6] showed that this compactiflcation is
ultrafilter coincides with its own closure.
T3
Indeed,
iff each non-convergent
Gazlk’s condition is also necessary
and sufficient in order for Richardson’s compactlflcatlon of a completely regular
topological space to be equivalent to the topological
Stone-ech
In 1972, the authors showed that a convergence space
a compact
T3
convergence space iff
X
a completely regular topological space.
completely regular.
X
compactlflcatlon.
can be embedded in
has the same ultrafilter convergence as
Such convergence spaces are said to be
Each completely regular convergence space has a
T
3
compac-
tlflcatlon with the same universal property which characterizes the topological
COMPACTIFICATIONS OF CONVERGENCE SPACES
Stone-ech
compactification.
347
Other formulations and proofs of essentially the same
results were given independently in 1973 by C. H. Cook [3] and A. Cochran and
R. Trail [2].
C. J. M. Rao [16], [17], and Vinod Kumar [24] investigated the conditions
X has a largest
under which a space
tion.
T2
T2
and smallest
and
T3
compactifica-
The summary of their results, along with some additions by the present
authors, is the subject of Section 3.
In 1971, Ellen Reed [19] made a detailed study of Cauchy space completions.
Motivated by Kowalsky’s completion theory [12], she defined "relatively diagonal"
and "relatively round" completions; in a later paper on proximity convergence
spaces [20], she introduced a relatively round compactification called the
"Y.-compactification".
Compactifications satisfying these two conditions receive
further attention in Sections 4 and 5 of this paper.
There are a number of directions from which the subject of compactifications
can be approached.
A recent paper by R. A. Herrman [7] gives a non-standard
development of convergence space compactifications. Another possibility is to
consider embeddings into spaces which are, in some sense, approximately compact;
this technique is used in
[8], [ii], and [18].
Our approach, like that of Rao, is to study the conditions under which a
space will have a largest and smallest compactification subject to certain conditions
(including the aforementioned properties introduced by Reed) imposed on the
compactification.
2
Some of our main results are summarized at the end of the paper.
PRELIMINARIES
Let
F(X)
denote the set of all filters on a set
will be abbreviated
The term "ultrafilter"
"u.f."; the fixed u.f. generated by x is denoted
A convergence space
X
X.
(X, /)
is a set
subject to the following conditions:
X
and a relation
/
between
F(X)
and
348
D.C. KENT arid G. D. RICHARDSON
(C I)
For each
(C 2)
If
(C 3)
If
F
x e X,
+ x.
/
x
and
G > F,
then
G
/
x
and
G
then
FDG
/
x,
x.
(X, /)
Ordinarily, a convergence space
term
/
/
x.
X;
will be denoted only by the
the
"space" will always mean "convergence space".
A space is
is made
T2
if each filter converges to at most one point; the assumption
throughout this paper that all spaces are
A space X
T3
is
closure operator for
containing
/
x
whenever
A
A subset
X.
Ux(X)
X
F
/
X
then
is
denotes the
X
compact if each u.f.
is said to be locally compact.
denote the
X
el
if every convergent u.f. contains a
X-neighborhood filter
at a point
the intersection of all filters which converge to
x e X,
unless otherwise indicated.
where
x,
of a space
A converges to a point in A;
compact set, then
Let
clxF
if
T2
is called a pretopological
x.
spac.e.
pretopological (topological) space coarser than
Ux(X)
If
/
For any space
X
is denoted
Ux(X)
x e X;
x
is
for all
X,
the finest
wX(IX).
A
continuous function will be called a map.
A compactification
and an embedding map
(Y,k)
K
k: X
/
Y
X
of a space
such that
ckX
consists of a compact space
Y
Y.
Since the term "compactiflcation" is used so frequently in this paper
we shall use the abbreviation
If
a map
i
f
(Yl’kl)
and
K2
"compr.".
(Y2’k2)
which makes the diagram
are compns, of a space
YI
X
commute, then
X,
and there is
i
is said to be
2’
then the two
2
larger than
2
(written
2 <- i )"
If
2
<
i
and
I
<
compns, are said to be equivalent.
In this section, we shall construct two compactifications which will play
a key role in the remainder of the paper; they are the one-point compatification
349
COMPACTIFICATIONS OF CONVERGENCE SPACES
and Richardson’s compactlflcatlon
Let
,
X be a space, and let
X
identity function from
*.
into
X [2 {a},
(i)
F
the restriction of
If
then
in
X;
(2)
X
to
X has no adherent point in
"one-polnt compactlflcatlons" for
X
X.
X,
to
converges to
#
x
x
one-polnt compactlflcatlon of X.
is called the
a
[
Let
be the
the finest convergence structure
assign to
subject to the following conditions:
the restriction of
where
F
F
X.
/
x
iff
in
+ a
iff
in
(,[)
Then
Although other non-equlvalent
is the only one that we
are possible,
shall consider in this paper.
Let
A___
N
X,
denote the set of all non-convergent u.f.
X
let
A*
X*
the filter on
{F e
A
Nx:A
H
>
F*; (2)
X
be the identity map of
PROPOSITION 2.1.
If
f:X
/
Y
If
If
G* be
H
+ x
/
y
in
in
X* Iff there is
X* Iff H >_
F
/
# *.
x in
Let
i*
The following proposition is proved in [21].
is a compact
X*
X
H
(X*,i*)
*
is a space, then
Y
is a map, and
such that the diagram
X*.
let
If
The convergence structure for
NX, then
into
X.
a space
G e F(X),
and for each
x e X, then
y
X
F},
{G*:G e G}.
generated by
X* is defined as follows: (i) If
X such that
e
’s on
T
3
is a compn, of
space, then there is a map
X.
f
commutes.
We now introduce some less familiar convergence space properties which turn out
to be important in the study of compns.
if every u.f. containing
A
A
A subset
of a space
converges to some point in
X;
X
is bounded
a space in which
every convergent filter contains a bounded set is said to be locally bounded.
The term "bounded" was suggested by Kasahara [9]
concepts is given by C. Riecke [23].
a further study of these
The notion of "boundedness" has been
studied by other authors under other names; for instance, H. Poppe [14] refers
to the same concept as "weakly relatively
compact".
D.C. KENT and G. D. RICHARDSON
350
A space
F v(/ {G
e
F
is said to be essentially bounded if, for each
X
NX:G
F})
#.
F I and F 2
mean that the filters
"F I
(In general, the statement
X
NX,
’’
2
v
contain disjoint sets.)
e
will
is said to be
essentially compact if NX is a finite set; this terminology is due to
Vinod-Kumar [24 ].
PROPOSITION 2.2.
A space
X
is essentially compact iff it is essentially
bounded and locally bounded.
PROOF.
An essentially compact space obviously has both properties.
Nx
other hand, if
F >_ / {G
such that
F
bounded; if
F
is an infinite set, then there is a free u.f.
NX:G # F}.
e
NX,
then
X
F
If
E
NX,
On the
on
X
X fails to be essentially
then
fails to be locally bounded.
Some additional characterizations of local boundedness and essential boundedness are given below.
PROPOSITION 2.3.
(i)
X
(2)
If
F
is convergent in
X,
then
(3)
If
F
is convergent in
X,
then
(4)
X
(5)
PROOF.
containing
F
The following statements about a space
F
is locally bounded.
is open in
F v(/Nx)
%,
X*.
< *.
(I) => (2).
F
If
F
/
(2) => (3).
F v GF.
assertion
X e F*.
(3) => (4).
then there is
X.
H
X,
H
>
such that each u.f.
Consequently,
,
is an u.f. containing
then
F
F
F* for some
F,
F*
F,
then for each F
But then, for each F
If
in
,
F v(f]NX)
If
such that
x
x,
converges to a point in
F*.
=> X
to a point
X are equivalent.
there is
and so
GF e Nx
GF
F*,
X*
X and converging in
filter
F
contrary to the
/
x
in
X.
X*
It follows
COMPACTIFICATIONS OF CObF@ERGENCE SPACES
.
Fv(ONx)
that
,
(4) --> (5).
a E
onto
The canonical map from
is clearly continuous if
(5) => (1).
If
X
X which converges to
X*
(i)
X
(2)
If
(3)
X*
is
F*
F
F
F
NX,
X
F,
containing
X*
(3)
(I).
X*,
in
X
in
X*
into
E*.
X
are equivalent.
*.
X U {F}
F
If
F
N
X
X
and
F
Since
F
1
(Yl,kl)
N
NX
X J{F}
implies
e N
is such that
X
iff
e(G)
/
YI
assuming that
Furthermore, if
8(y)
and
in
X*
X*.
,
to
F
F v(D{G
X*
is the only filter
e X*
e
Thus
X.
,
NX:G # F))
then
F
X which converges to
is a compn, of a space
X*.
F /
X*
F.
containing
X
F*.
F*
X which can converge in
If
is essentially bounded, then there
is the only member of
which implies
one-to-one and onto
YI
x
Thus there is an u.f. containing
X
is easy to see that in the commutative diagram
in
/
and the canonical map from
one can construct a free u.f. containing
E
F
is discrete.
(2) => (3).
If
is a closed set.
F.
E
Therefore,
then
such that
ffi>
F
X
X*
which carries
is essentially bounded.
{}
F
X
,
The following statements about a space
(I) => (2).
PROOF.
X*
for all
x
fails to be continuous.
PROPOSITION 2.4.
into
is not locally bounded, then there is
F* /(X*- X) # $
such that
X*
351
G
such that
k
I
X
)Y
E
1
>_ E*,
then it
e
the map
is an u.f. on
YI’
X*.
in
then
is
G
/
y
Thus, there is no loss of generality in
have the same underlying set, an assumption which
we shall use whenever convenient.
Finally, if
E
(Y,k)
shall consistently denote by
is a compn, of
X
the function
such that
:X*
/
Y
E* > E,
defined by
then we
#(a)
y
D.C. KENT and G. D. RICHARDSON
352
if there is
P
a
/
X*
in
P
X
such that
k(i*) -I
and
P
/
y
in
Y.
$
is
always well-defined but not necessarily continuous; in later sections we shall
3.
X* to Y.
as the canonical function from
refer to
2 AND
T 3 COMPACTIFICATIONS.
eek
In this section we
to clarify, simplify, and extend results initially
[24].
obtained in [16], [17], [22], and
It should be noted that, for all of the
(C 3)
results of this section, the convergence space axiom
can be replaced by
the weaker axiom:
F
(C)
LEMMA 3.1.
(meaning
c1
f:X
X),
THEOREM 3.2.
/
Y
implies
x
and the restrlctlon of
has a smallest compn.
(2)
X
is open in each of its compns.
(3)
X
is essentially compact.
(4)
X has a largest compn.
ffi>
Assume that N
Y
and with the property that
to some fixed point
i*(X)
f
is a homeomorphlsm,
A
to
is an infinite set.
X
equipped with a convergence structure which
X,
A a dense subset of X
spaces,
x0
in
X are equivalent.
(i) &d (2) was established by Rao
The equivalence o5
(3).
x.
The following statements about a space
X
(2)
/
is a map between
(I)
PROOF.
F
f(A).
A) CY
f(X
then
If
/
Let
agrees with X*
Y
X
in
[16].
t
N
X
be
on filters containing
every fee u.f. which contains Nx converges
X.
Then
X,
is a compn, of
(Y,i*)
in
and
X is clearly not open in Y.
(3) => (4).
It is easy Co verify that
-
(X*,J)
is the largest compn, of an
essentially compact space.
(4) => (2).
Let
(Y,k)
Leu 3.1 and the fact that
be the largest
: < :,
T2
it follows that
compn, of
k(X)
X.
Using
ts an open subset
353
COMPACTIFICATIONS OF CONVERGENCE SPACES
Y.
of
(Z,g)
Let
T2
be any
X;
compn, of
then there is a map
f
which
makes the following diagram commute.
x- Y
* <_ g,
Since
it is easy to verify that
of Lemma 3.1 and the fact that
f(Y
kX)
Z
g(X)
Y
k(X)
f
is closed
Rao’s proof
in
onto
Z.
Making use
and, consequently, compact,
is also compact, and therefore
Vinod-Kumar [24] questioned
Y
must map
g(X)
is open in
Z.
[17] of the equivalence of
statements (3) and (4) of Theorem 3.2; neither noticed the equivalence of
Btatements (2) and (3).
largest and
For an essentially compact space
X, <*
is the
is the smallest compn.
A space is defined to be completely regular if it is
T
and has the same
3
ultrafilter convergence as a completely regular topological space.
The next
theorem is proved in [22].
THEOREM 3.3.
A space
X
has a largest
T3
compn, iff
X
is completely
regular.
The largest
T3
compn, is constructed by making relatively minor modifi-
cattOnB in the convergence of filters relative
compn.
details can be found in
to the topological Stone-Cech
[22].
In the final theorem of this section, we add two alternate characterizations
that given by Rao for spaces having a smallest
THEOREM 3.4.
T3
compn.
The following statements about a completely regular convergence
space are equivalent.
(I)
X has a smallest
(2)
X
(3)
X
is a locally compact convergence space.
(4)
X
is open in each of its regular compns.
T3
compn.
is a locally compact topological space.
D.C. KENT and G. D. RICHARDSON
354
The equivalence of (I) and (2) was proved by Rao in [16].
PROOF.
(2)
(3).
ffi>
X
Since
X have the same u.f. convergence, they have
and
the same compact sets, and so every
(3)
ffi>
u.f. on
Z
G>
F.
clzk
k(X),
(4).
(Z,k)
If
X
Since
ffi>
(2).
then there is an u.f.
kX,
k(X)
is open in
X
Since
X,
is any regular compn, of
is locally compact,
and therefore
(4)
Z
containing
X-convergent filter contains a compact set.
G
F
and
on
X
G
is any
such that
cannot converge to a point in
Z.
(4) implies that
is completely regular,
X
open in each of its compactifications; since
is a topology,
X
X
is
is locally
compact.
If
X
is a space satisfying any of the equivalent conditions of Theorem
is the smallest
then
T
3
compn.
RELATIVELY DIAGONAL AND RELATIVELY T
4.
3.4,
3
COMPACTIFICATIONS.
In the preceding section, we studied compns, subject to convergence space
3. In this section and the next, we deal with properties
of the compns, themselves; these properties are not meaningful when applied to
T2
properties
T
and
The concept of a strict compn, was introduced in [i0],
the underlying space.
T3
where it was shown that the strict
X
compns, of a completely regular space
correspond in a one-to-one manner with a certain class of Cauchy structures
compatible with
X.
introduced by Reed
Relatively diagonal and relative round compns, were
[19] as Cauchy space completion
properties, and relatively
T
is a new compn, property which is being introduced here for the first time.
3
Before formally defining these terms, some additional notation is needed.
-
(Y,k)
Let
u
is a function
(y)
If
if
y
denote a compn, of a space
c: Y
/
k(X).
[], A C Y,
and
FC/)
Let
F
such that
[]
u(Y)
X.
/
A
Y in Y,.
denote the set of all
F(Y),
then let:
selection function
_>
c(y), and
selectlon functions,
COMPACTIFICATIONS OF CONVERGENCE SPACES
A
{y
F
{A C Y:A
(y))
Y:A
It is easy to see that
A
OC
A and
Under the assumptions of the preceding paragraph, let
Y,
[K],
u
or else
Y
F <
such that
Again, let
pA
the
A < B
and define
A
355
o(y).
F
If
AC B
to mean
e F(Y)
and
A,B be subsets of
and, for all
u
[K I
X.
If
define
Y, B
y
r
o(y)
F
A).
(Y,k)
K
tl(cA kX)
form pF for F
A
be a compn, of
pF
and let
F.
A CY
be the filter on
and
Y
F e F(Y),
define
generated by sets of
(We denote these concepts by pA and
respectively, if there is no possibility of confusion regarding the intended
eompn. )
For the purpose of formulatlng the following four definitions, we continue
(Y,k)
assuming that
DEFINITION 4.1.
there is
G
is a strlet compn, of
is a relatlvely d.i..agonal compn, of
Y
implies
F
/
y
in
r
F
/
y
/
k(X)
/
y
in
Y whenever
/
is a relatively T
y in Y whenever
/
y
in
y
in
X
X
X
Y.
PROOF.
(Y,k)
Y0
Y.
in
Let
a filter
if
y
v:Y
/
be a relatively dlagonal compn, of
F(X)
(y) e F(X)
k(x)
for
if, for each
is strict and
if
Each relatively diagonal compn, is strict.
K
Y,
if, for each
PROPOSITION 4.5
Let
in
Y.
compn, of
3
y
Y.
is a relatively round compn, of
in
F
c G.
K
y e Y,
V(Y)
G and F >__
DEFINITION 4.2.
y
DEFINITION 4.4
/
if, whenever
such that
[K]
F
X
Y
/
DEFINITION 4.3.
F
X.
in
[],
p
is a compn, of
X,
and let
be a function which associates, with each
such that
x e X.
Given
@
> (y), k((y)) + y in
A X,
define
A
Y
and
{y e Y:A e V(y)),
D.C. KENT and G. D. RICHARDSON
356
G
and let
k(B(y)) #,
(y)
F >_
show that
Fu
E
LEMMA 4.6.
If
[<],
(pF)
PROOF.
F e
F
F
The assertion
F
is an u.f.
A.
A
F
)C A C A.
K
Y
on
(cF-k(X))
be defined by
F.
Since
Y,
in
YO
is a relatively diagonal compn.,
K
is a strict compn.
and
Is a compn, of a space
F
<
f
F
y
Then
If
and thus
u
r
and
(y)
F e
K and K
A e (y)
In Y.
+ y
us
F <
A
u(y)
is establlshed
F
F
y e g- k(X),
for
and
Then there is
F
and thus there is
g- F
F(Y),
and so
<
r
E
F.
u(y),
F
F
A
pF
A
Y
then
A
Let
is obvious.
F
X,
.
<
F
<
such that
A
[<]
e
By straightforward arguments one can
Consequently
(pF)u.
A
F11y let
y e
F <
such that
(cF-
k G-
< r
which mplles
F
kG >
(Y,k)
<
Y.
y
and
Therefore
then
Finally, let
for all
ck G,
Y.
in
YO
/
e F}.
{A C X:A
such hat
then there
is would imply
A,
mplyng
A
r
,
and the proof is complete.
THEOREM 4.7.
(2)
T
3
(I) A relatively round compactlflcatlon
is relatively diagonal.
A relatively T 3, relatively diagonal compn, is relatively round. 3) .A
compn, is relatively T
3.
PROOF. The first two assertions follow immediately from Lemma 4.6; the
third is obvious.
THEOREM 4.8.
and relatively
(i)
For any space
X,
the compn.
*
is relatively round
T 3.
(2)
For any space
X,
(3)
The compn.
of a space
K
the compn.
X
is relatively round.
is relatively
T3
iff
X
is locally
bounded.
(I)
PROOF.
set
A
X.
(me can routinely vertify
that
From this result it follows that
implies strict)
and also relatively
T 3.
A*
*
p(A*)
(A*) u
for any
is relatively diagonal (which
It then follows by Theorem 4.7 that
357
COMPACTIFICATIONS OF CONVERGENCE SPACES
K*
(2)
F
(3)
>
’
[],
u e
For each
filter
F
.,
is also relatively round.
one can routinely verify that
r
F
F
for each
which converges in
F
If
Nx.
F(X),
e
G
if
/
x
X,
in
G / A.
p(G)
then
in
a
/
F
u.f., and
an
p(F)
A >
then
iff
Thus the assertion follows
by Proposition 2.3.
THEOREM 4.9.
A space
has a largest strict, relatively diagonal, or
X
relatively round compn, iff
X
is essentially compact.
K*.
largest compn., if it exists is equivalent to
PROOF.
X
If
is essentially compact, then
Conversely, assume that
*
Since
<*
is known to be the largest
and the desired conclusion follows by theorem 4.8.
X,
compn, of
In each case the
*
is strict,
<
K
,
(Y,k)
X.
is the largest strict compn, of
and in accordance with our remarks at the end
of Section 2, we shall assume that
K*
and
have the same underlying set
K
and the same convergence relative to u.f.’s.
From the fact that
k
is strict
(indeed, relatively round), and Proposition 2.3, it follows that
X
must be
locally bounded.
Next, assume that
F
such that
({{G
v
identity map from
which makes
J(F)
that
/
a
<’
in
’
Z
X
X
e
and
NX:G # F})
Z,
into
(Z,j)
.
is not essentially bounded.
#
X
/
b
in
Z
from
into
for
Z
Z
J
let
be the
a convergence structure
G e Nx and G
X,
F.
One can show
and one can show that the
is not continuous.
This argument
shows that the existence of a largest strict compn, also requires that
be essentially bounded.
Since we showed earlier
it follows by Proposition 2.2 that
x
subject to the conditions:
is a relatively round compn, on
canonical function
X {a,b};
Z
and assign to
a compn, of
j(G)
Let
e N
Then there is
X
X has to be locally bounded,
X must be essentially compact.
This,
358
D.C. KENT and G. D. RICHARDSON
K* <
along with
,
K*.
implies that
<
is equivalent to
Had we begun by assuming that
<
is the largest relatively round or
relatively diagonal compn., precisely the same argument can be used to con-
clude that
X
is essentially compact and
THEOREM 4.10.
If
The following statements about a space
(I)
X
(2)
X has a smallest strict compn.
(3)
X has a smallest relatively diagonal compn.
(4)
X
(5)
X has a smallest relatively regular compn.
X
has a smallest relatively round
(I) => (2).
PROOF.
e
of
X.
:Y
be a filter
F >_
cG
it is necessary
is continuous for any strict compn.
will be continuous if there is no
/
y
in
Y
k(X).
y
for some
existed, then by the assumption of strictness there would
G e F(Y)
such that
X
the assuption that
so
+
It is clear that
F
X,
is a strict compn, of
Since
F(Y), where Y- k(X) e F and F
If such a filter
oomph.
.
only to show that the natural map
F
are equivalent:
bounded, then the smallest compn, subject to each of the
specified conditions is
u.f.
X
is locally bounded.
is locally
(Y,k)
K*.
equivalent to
G- y
in
Y,
k(X) e G,
is locally bounded guarantees that
is impossible.
Thus
is the smallest strict compn, of
0:Y
/
X
and
F
k(X) e
>
cG.
clyG,
But
and
(,i)
is continuous, and
X.
The same argument is valid if "strict" is replaced by "relatively round",
"relatively regular", or "relatively diagonal".
implies conditions
(2) => (i).
be equivalent to
is continuous iff
Thus condition (I) also
(3), (4), and (5).
.
It is easy to show that the smallest strict compn, of
X
By Proposition 2.3, the canonical map
is locally bounded.
Since
K*
of
X*
X must
on
is also a strict compn.
COMPACTIFICATIONS OF CON%rERGENCE SPACES
This argument is also valid if "strict"
X must be locally bounded.
X,
of
359
is replaced by "relatively diagonal" or "relatively
round".
Thus conditions
(3) and (4) also imply condition (i).
(5) => (i).
is not locally bounded, and let
X
Assume that
X
denote a relatively regular compn, of
such that
!
*-
Then
K-
(Y,k)
Y
k(X)
X*
must be an infinite set (otherwise, continuity of the canonical map from
Y
onto
Z
and let
and
Yl
Let
would be violated).
be the quotient space derived from
’ ’. (Z,k)
Then
Y2"
X
compn, when
by identifying the points
X
is also a relatively regular compn, of
and
can have no smallest relatively regular
X
Thus
it is clear that
Y
k(X),
Y
be arbitrary points in
YI’ Y2
is not locally compact.
We next consider some lifting properties of certain types of maps relative
to relatively
relatively diagonal, and relatively round compns.
T3,
However
we first need some additional terminology.
(Y,k)
Let
of all filters
F
e
F(X)
such that
<-Cauch7 structure for
the
If
X
I
C<2
THEOREM 4.11.
are relatlvely
unique map
F
for each
f
T
If
k
3
k
I
Y2
f
1
/
2
C
converges in Y.
/
X
X
2
is said to be a
is a
2
X1
KI (Yl’kl)
and
Kl2-Cauchy
X 2,
denote the set
C
is called
-Cauchy filters.
K2
(Y2’k2)’
map
if
and
Kl2-Cauchy
map, where
<1
and
K2
respectlvely, then there is a
which makes the following diagram commute.
X2
YI
CI.
f:X
compns, of
f
1
I
and let
and its members are called
X,
f:X
respectively, then a map
f(F) e
k(F)
are spaces with compns.,
X2
and
X,
be a compn, of a space
D.C. KENT and G. D. RICHARDSON
360
For each
PROOF.
(y)
and define
a
YI’
Y
k2f()
where
z,
CI
G
choose
z
/
y
YI’
in
f
It remains to show that
is a well-deflned function.
show that
/
f
is
are sufficient to
C 3,
map, along with convergence space axiom
l2-Cauchy
G
I
The assumption that
Y2"
in
k
such that
is
cont Inuous.
F
Let
and
in
/
y
klXl F2"
gl
G, F 2 >_
klXI
Let
cIG,
F 2.
kl X
(O)
/
/
is a relatively
2
(plG) >_ pK2(G).
it remains only to show that
(F2)
and
z
z
Thus
Y2"
in
F 2 >_
Note that
gl"
in
where
2,
(F I)
G e F(Y I)
is strict, there is
y
/
IF
It is immediate that
z.
KI
G
F
F
in the form
(y)
and
Since
and it follows that
F
Write
Using the fact that
Y2"
gl
YI"
in
k
2
kl-I (FI)
f
such that
p<lG,
T
klXI F I
since
compn, of
3
X 2,
But this is easily established,
(F)
/
z
and the proof
Y2’
in
is complete.
then
i:X
/
X
is a
COROLLARY 4.12.
of
X,
any compn, of
is a space,
X
If
K*K
Cauchy map.
For any space
and
the identity map on
i
X,
Thus we obtain
is the largest relatively
*
X,
T
3
compn.
X.
Theorem 4.11 would not, in general, be a correct statement if "relatively
T3"
were replaced by "relatively round", "relatively diagonal", or "strict"; otherwise,
there would always be a largest compn, of any space
X
subject to these pro-
However the lifting theorem that follows
perties, contrary to Theorem 4.9.
applies to relatively diagonal and relatively round compns, as well as relatively
T3
compns; it generalizes the lifting property of the compn.
THEOREM 4.13.
Let
X
relatively round, or relatively
f:X
/
F
C
Z
(Y,k)
have a compn.
T 3.
a map with the property that
Then there is a unique map
Let
f(F)
Z
*.
which is relatively diagonal,
be a compact
space, and
T3
is convergent in
Z
for each
which makes the following diagram commute.
z
--
COMPACTIFICATIONS OF CONVERGENCE SPACES
361
k
X
PROOF.
Y
is equivalent (in the
(Y,k)
is relatively dlagonal, then
If
sense defined in [19]) to some member of the family of Cauchy space completions
of
(x,C k)
Thus, for relatively diagonal and relatively
defined in [19].
round compns., the assertion follows from Theorem 4 of [19].
If
is relatively
relatively
T
3)
’
compn.
Z
then we can regard
T3,
of itself, where
C,
(and hence
3
Z-convergent
is the set of all
f
Then the assumptions of the theorem imply that
filters.
T
as a
K’
is a
Cauchy
map, and the conclusion follows as a corollary to Theorem 4.11.
We conclude this section by showing that relatively round compns, need not
Indeed, it follows by Theorem 4.8 that
be relatively
T3,
for any space
X which is not locally bounded,
and vice versa.
T
of a space
X which is not relatively diagonal.
3.
EXAMPLE 4.14.
compn.
be the unit interval [0,i] of the real llne with its
I
Let
(an
Let
usual topology.
H
T3
In the example that follows, we construct a strict
relatively
let
is relatively round but not
be a sequence in
I
be the filter on
[0,I] which converges to
(an).
generated by the sequence
Let
Y
0
I;
in
be the
space consisting of the set [0,I] with convergence defined as follows:
F
y # 0,
For
(i)
y
/
/ H.
F >_
CllG
[0,1]
{a :n-- 1, 2,...},
(i)
s(x)
A e Hs
X
(Y,i)
it follows that
Let
If
R for x e X;
A
/
y
in
i
(2)
F
0
in
converging to
is the subspace of
and
I;
Y
0
/
I
T
3
Y
iff there
such that
determined by the subset
the identity embedding of
is a strict
in
compn, of
X
Into
Y,
X.
be the selection function defined as follows:
s e []
then
F
iff
GI,...,Gn
u.f.’s
is a finite set of
Y
in
s
e
H,
(2)
Uy(an)
S(an)
and therefore
A
s
for
n
1,2,
If
contains all but finitely many of
then
D.C. KENT and G. D. RICHARDSON
362
the
a
n
’s.
0
converge to
so
H
It follows that
Y,
in
is not finer than any of the filters which
H 0
and therefore
in
H
But
Y.
0
/
Y,
in
and
is not relatively diagonal.
SIMPLE COMPACTIFICATIONS.
5.
A compn.
(Y,k)
<
y e Y
and, for each
(Y,k)
X
of a space
X,
is said to be simple if
Uy(y)
the neighborhood filter
Y
/
y
<
in
is strict
Y.
A strict
is a pretopologlcal
compn.
<
space.
Note that only pretopologlcal spaces can have pretopologlcal compns.
will be called pretopologlcal if
We omit the straightforward proof of the first
The following statements hold for any (pretopologlcal)
PROPOSITION 5.1.
space
X.
(i)
*
(2)
<
(pretopologlcal) Iff
is simple
PROOF.
H
Let
A simple,
(Y,k)
<
F(Y),
e
(pretopologlcal).
is simple
PROPOSITION 5.2.
let
H
y e
< p
H. Let A
cl
k(X),
Uy(y)
(y)
A e (y).
<
e r
COROLY 5.3.
round compn, of
[<].
e
(y)
H; then
<
Since
Uy(y)
for
3.
F
and
H
(cH-
k(X))
For any space
X,
K*
k(X).
y e Y
H
e F(Y)
H <
such that
such that
H <
since
A,
We shall show that
A,
H e
A.
If
F,
and
it follows that
and the proof Is complete.
Is the largest sple, relatively
X.
This is an immediate consequence of Corollary 4.12 and Propositions
PROOF.
5.1 and 5.2.
For simple compns., the converse of Proposition 5.2 does not hold.
compn.
round.
<
X,
is simple, we can assume without
H e
there is
(y)
Thus Y- H
Consequently,
T
relatively round compn, is relatively
then there is an u.f.
F
is locally bounded.
X
be a simple, relatively round compn, of a space
and let
loss of generality that
r
proposition.
constructed in Example 4.14 is simple and
T3,
The
but not relatively
363
COMPACTIFICATIONS OF CONVERGENCE SPACES
A pretopological compn.
V
PROPOSITION 5.4.
Then
Let
Let
PROOF.
it is sufficient to show that
diagonal
[K].
and
y e k(X),
If
for each
y e Y
y e Y
k(X),
If
V which is
any set
for
K-basic
and the desired equality is established.
Conversely, assume that
by
(Uy(y))o,
Uy(y)
or
V
V
X.
is relatively
To show that
this assertion is obvious.
then it is easy to check that
y,
y.
is relatively diagonal.
K
be relatively topological.
K
V
implies
Uy(y)
Uy(Z).
be a pretopological compn, of a space
is relatively topological iff
K
k(X))
basic sets for
K
(Y,k)
K
there is a base of sets for
V f (Y
z
of this type will be called
V
Sets
such that
is said to be relatively
X
of
k(X),
Y
topological if, for each y
consisting of sets
(Y,k)
K
Uy(y)
(Uy(y))
o(y)
Uy(y)
#
y e Y
k(X)
for all
y e Y,
and sets of the form
k(X).
y e Y
and
y e k(X).
for
{V:V
Then
Uy(y)}
are
is relatively topological.
Thus
X,
For any pretopological space
is the largest relatively
K*
X.
topological compn, of
PROOF.
o(y)
for all
K-basic for all points
THEOREM 5.5.
is relatively diagonal, and define
K
is relatively topological is an immediate
K*
The fact that
consequence of Theorems 4.7 and 4.8, along with Proposition 5.1 and 5.4.
Let
(Y,k)
K
Let
be the canonical function.
F
Then there is an u.f.
k(F)
/
y
Y;
in
then
completed by showing
If
X
F*,
then for each
X*
let
(F*)
F,
(F*)
be the point in
y
choose
(YF)F
generated by the net
z
-
Y
8 * a
such that
e
F*.
/
in
X*
by definition of
$,
and the proof will be
Vy(y)
>
Suppose that
clearly follows.
F" Let K
to which
and
Y.
F*- X,
F
/
(K)
and let
H
converges.
X
be the filter on
be an u.f. finer than
If
V
is a
H,
Y
X*.
Assume
in
in
a
F*
:X*
and let
a
such that
(a)
X*
be an u.f. on
8
X
on
y
then
F
X,
be a relatively topological compn, of
and
K-basic
D.C. KENT and G. D. RICHARDSON
364
Since
k(F)
hence
y
Y
is an u.f. and
z.
(K)
Therefore,
y
/
Y,
in
(F*)
and therefore
PROPOSITION 5.6.
X of Y,
Y
the subspace
y
/
is a relatively
Let
PROOF.
Y
ClzA
such that
Assume that
ClzK
that
.
A C Z
t
Let
t
in
Y.
in
Y.
Thus
/
/
and let
K
y
E
Y.
in
to
Z
If
and
is,
is a regular topological space.
cIA.
Z e
y,
y
/
X
compn, of
3
and
K
/
H
Then there is an u.f.
A
be an u.f. containing
Since
t
T
is a topological space.
Z
in addition, a simple compn., then
(H)
and
Y.
in
Z
then
Y,
in
F* converges
of any u.f. finer than
(Y,k)
If
z
and, consequently,
Y
in
/
F.
F
for all
k(F)
is pretopologlcal,
It follows that the image under
y
k(F) / V
then one can show that
z,
neighborhood of
K
and
y,
y
cl
such that
T 3,
is relatively
clzA.
y
/
in
! H.
Z
it follows
Since the closure
is also
operator for
Z
is idempotent,
simple, then
Z
is a pretopological, and hence topological, space; the regularity
of
Z
Z
is a topological space.
is an easy consequence of the assumption that
THEOREM 5.7.
A (pretopological) space
gical) compn, iff
X
is locally bounded.
compn., when it exists, is equivalent to
PROOF.
.
X
If
<
is relatively
T
3.
has a smallest simple (pretopolo-
The smallest simple (pretopological)
The argument used to establish the equivalence of Conditions (I)
and (5) in the proof of Theorem 4.10 can be applied to establish this result.
We next turn to the problem of characterizing those spaces having a largest
simple or pretopological compn.
For the former property, the problem has not
yet been solved in its full generality.
A property slightly weaker than essential
compactness is needed for the solution of the problem; this property is defined
and discussed in the next paragraph.
A space
X
is defined to be almost
most one point in
in
X*.
X*
essentially compact if there is at
to which a free filter containing
X*
X
converges
This property can a,lso be characterized internally, albeit more clumsily,
COMPACTIFICATIONS OF CONVERGENCE SPACES
X
as follows:
NX:G
v (/9{G e
,
F})
F
THEOREM 5.8.
If
simple compn, of
X.
X
canonical function.
X
F
such that, for some
/
F v(/]NX)
x,
X,
G
:X*
X e G.
X*
establish continuity of
#,
it is sufficient to show that
X*
such that
(G)
k(X) e
Y*
is a free u.f., then
it is clear from the conditions imposed on
Y
Suppose, on the other hand, that
F*.
>
each
X*
in
F,
F e
G e G
{k(HF):F
y
and so
X
F})
e
(G)
z;
z
/
to
Y.
in
;
is the only point in
z
such that
{y};
HF
Since
+
such that
#(b)
Let
y.
e
<
But
k(F)
/
b
in
Then
Now
X*
y e Y
by Lemma 3.1,
F*) G.
k(X).
y e Y
for some
e F(X)
F e F.
k({HF:F
(HF)
and
k(X).
y,
For
which
is simple,
F
>/{HF:F
e F}
implies
kF
/
y,
z.
THEOREM 5.9.
If
the
by construction of
that
()
such that
X
for all
+ y
e F}
e N
X
(G)
such that
HF
choose
k(HF)
implies
F
implies there is
Choose
Y
k(X) can converge.
Y
to which a free u.f. containing
b
/
b e X*
in
(G)
is the largest
is essentially compact, the conclusion follows
+ b
If
*
and
by Theorem 3.2, so assume that there is exactly one point
there is an u.f.
.
is essentially bounded, and there
be a simple compn, of
X
If
is locally
has the property that
X
is almost essentially compact, the
(Y,k)
Let
N
of
or else
x e X
is at most one point
PROOF.
X
is almost essentially compact iff, either
bounded and at most one member
F
365
Let
X
be a space which is locally bounded (pretopological).
has a largest simple (pretoplogical) compn., then
X
is almost essentially
compact.
PROOF.
X
If
is not almost essentially compact, then there are at least
two distinct points
converges to
are in
X*
a
X.
and
a, b
b.
in
If
Thus, since
X*
X
*
such that free fiters containing
is locally bounded, then necessarily
is simple, the assumption that
X
X*
a
X
and
is either
b
366
D.C. KENT and G. D. RICHARDSON
Ux,(a)
pretopological or locally bounded leads to the conclusion that
and
Vx,(b)
Note
that
Choose
b.
/
Ux,(a)
A e
A- (X U {a})
AI
and
B
Ux,(b)
B e
and
AB
such that
(X ] {b})
B
I
+ a
.
are both infinite sets;
AI
with no loss of generality, assume that the cardinality of
does not exceed
B I.
that of
both consist of free u.f.’s on X. Let the members of A
I
I and B I
{F : I}; then under our cardinality assumption,
be indexed as follows: A
I
Now A
we can index a subset
Finally, we define a
(i)
AI
{G : e I}
same index set I.
I with the
totally bounded Cauchy structure C on X consisting of:
B2
or
B2;
Let
Y
(3)
e N
(A
X
(2)
all members of
C
be the set of
It B2)
(3)
([])
(i)
([Fu
/
injection function given by
j(x)
Cauchy structure denoted by
CF
straightforwardly that
$:X*
/
Y
Let
equivalent classes.
Ga])
;
Ga,
F
the Cauchy equivalence class determined by
and
N
not included in
X
FG
all filters finer then filters of the form
function defined as follows:
F
X;
all convergent filters on
B
of
(Y,J)
is not continuous.
[].
in
[19],
(2)
all
e C.)
where
*
a largest simple (pretopological) compn, of
u([F])
F
e I.
(Here,
Let
j :X
/
be the
if
Y
[F]
denotes
be the natural
F
{0},
then one can verify
(pretopological) compn, of X,
is a simple
Since
F(X)
/
I.
is equipped with the complete
Y
If
:Y
for
is the only possible candidate for
X,
it follows that
X
has no largest
simple (pretopological) compn.
COROLLARY 5.10.
compn, iff
X
is almost essentially compact.
COROLLARY 5.11.
iff
6.
X
A pretopological space X has a largest pretopological
A locally bounded space
X
has a largest simple compn.
is almost essentially compact.
SUMMARY.
A space X has a largest compn, iff
X
is essentially compact.
The same
condition is also necessary and sufficient for the existence of a largest strict,
COMPACTIFICATIONS OF CONVERGENCE SPACES
relatively diagonal, or relatively round compn.
X
If
367
is pretopological
(locally bounded), then X has a largest pretopological (simple) compn, iff
X
is almost essentially compact.
compn, and a largest simple
Every space
compn., when it exists, is equivalent to
X
T
has a largest relatively
relatively round compn.
has a largest relatively topological compn.
A space
X
3
Every pretopological space
In every case cited, the largest
K*.
has a smallest compn, iff
X
is essentially compact
A weaker
condition, local boundedness, is necessary and sufficient for the existence of
a smallest compn, subject to each of the following properties:
diagonal, relatively round, relatively
T3,
and simple.
strict, relatively
Local boundedness is
also necessary and sufficient in order for a pretopological space to have a
.
smallest pretopological compn.
it exists, is equivalent to
In each case cited, the smallest compn., when
REFERENCES
"Convergences", Annales
Univ. Grenoble 23 (1948) 57
112.
i.
Choquet, G.,
2.
Cochran, A. C. and Trail, R. B., "Regularity and Complete Regularity for
Convergence Spaces", Proc. of VPI Topology Conference, Springer Lecture
Note Series No. 375 (1973) 64- 70.
3.
Cook, C. H., "Compact Pseudo-Convergence", Math. Ann. 202 (1973), 193
4.
Fischer, H. R., "Limesraume", Math. Ann. 137 (1959), 269
5.
Frechet, M., "Sur quelques points du calcul fonctionnel", Thse, Paris 1906,
et Rend. Circ. Mat. Palermo 22 (1906).
6.
Gazik, R. J., Regularity of Richardson’s Compactification", Can. J. Math. 26
(1974), 1289 1293.
7.
Herrmann, R. A., "A nonstandard Approach to Pseudotopological Compactifications",
Notices A.M.S. 26 (1979) AI21. Abstract #763-54-3.
8
Hong, S
9.
Kasahara, S., "Boundedness in Convergence Spaces", Proc. Reno Conference on
Convergence Spaces (1976), 83- 92.
S
and Nel, L
D
202.
303.
"E-Compact Convergence Spaces and E-Filters"
D.C. KENT and G. D. RICHARDSON
368
REFERENCES
i0.
Kent, D. C., and Richardson, G. D., "Regular Completions of Cauchy Spaces",
Pacific J. Math. 51 (1974), 483- 490.
ii.
Kent, D. C., Richardson, G. D., and Gazik, R. J., "T-regular-closed Convergence
Spaces", Proc. Amer. Math. Soc. 51 (1975).
12.
Kowalsky, H. J., "Limesrume and Komplettlerung:, Math. Nachr. 12 (1954),
301
340.
13.
Novak, J., "On Convergence Spaces and Their Sequential Envelopes", Czech.
i00.
Math. J. 15 (90) (1965), 74
14.
Poppe, H., Compactness in General Function Spaces, VEB Deutscher Verlag der
Wissenschaften, Berlin, 1974.
15.
Ramaley, J. F., and Wyler, 0., "Cauchy Spaces II. Regular Completions and
199.
Compactiflcations", Math. Ann. 187 (1970), 187
16
Rao
C J. M
"On Smallest Hausdorff Compactlflcatlon for Convergence Spaces"
230.
Proc. Amer. Math. Soc. 44 (1974), 225
17.
"On Largest Hausdorff Compactlflcatlon for Convergence Spaces",
Bull. Austral. Math. Soc. 12 (1975), 73- 79.
18.
"On m-Ultracompactlflcatlons", Proc. Reno Conference on
187.
Convergence Spaces", (1976), 184
19.
20.
Reed, E. E., "Completions of Uniform Convergence Spaces", Math. Ann. 194 (1971)
83
108
"Proximity Convergence Structures", Pacific J. Math.
21
"A Stone-ech Compactlflcatlon for Limit Spaces"
Richardson, G D
Amer. Math. Soc. 25 (1970), 403- 404.
22.
Richardson, G. D., and Kent D. C., "Regular Compactifications of Convergence
Spaces", Proc. Amer. Math. Soc. 31 (1972), 571- 573.
23.
Riecke, C. V., "On Boundedness for Convergence Spaces", Bolletino U. M. I. (5)
15-B (1978), 49- 59.
24.
Vinod-Kumar, "On the Largest Hausdorff Compactification of a Hausdorff
Convergence Space", Bull. Austral. Math. Soc. 16 (1977), 189- 197.
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