I ntrn. J. Mh. Mah. S.
Vol. 4 No. 3 (1981) 451-472
451
THE STAR COMPACTIFICATION
G.D. RICHARDSON
Department of Mathematics
East Carolina University
Greenville, N. C. 27834
D.C. KENT
Department of Pure and Applied Mathematics
Washington State University
Pullman, Washington 99164
(Received September 18, 1980)
ABSTRACT.
The relationships between a convergence space and its star compactifi-
cation is studied.
Special attention is given to lifting properties of this
compactification.
In particular, it is shown that a natural extension of any
continuous function to the respective compactification spaces is
@-continuous.
KEY WORDS AND PHRASES. Convergence space, compactification, G-space, R-series,
natural extension, @-continuous function, proper map, open map.
1980 MATHEMATICS SUBJECT CLASSIFICATION CODES.
452
G.D. RICHARDSON and D.C. KENT
INTRODUCTION.
1.
We study a convergence space compactification which was introduced by one of
the authors in 1970 (see
[II]).
The star compactification m *
(X*,i*)
of a
convergence space X is constructed by adjoining to X the set X’ of all non-convergent ultrafilters on X and constructing a compact convergence structure on
X*
X U X’ in a natural way.
from a T
[II]
It is proved in
that a continuous function
space X into a compact T space Y has a continuous extension to
3
2
[7],
The authors published a survey paper,
X*.
concerning the existence of largest
and smallest convergence space compactifications relative to various constraints.
In all cases studied, the largest compactification, whenever it existed, turned
g:.
out to be
In a :re recent paper,
[9],
we showed that
characterize e-regular and completely regular spaces.
K*
can be used to
These results suggest that
further investigation of the star compactification is appropriate.
,
In Section 2, we examine the relationship between the decomposition series
of X and X
showing that the lengths of these series can differ by at most one.
These results yield a method for constructing compact T 2 spaces with arbitrarily
long decomposition series.
In Section 3, the R-series of X and
X*
are compared.
By means of the R-series, the notion of e-continuity and other e-mapping properties
(see
[2], [3])
are extended to convergence spaces.
If f is a function from a space X to a space Y, then a "natural extension"
f
T
2
X*
/
Y*
is defined in Section 4.
The natural extension is unique if Y is
and coincides with the continuous extension constructed in
compact and T
3.
[II]
when Y is
The main result of Section 4 is that any natural extension
f,
is e-continuous whenever f is continuous.
This result is used to obtain, among
other things, an alternate construction of
BX
Section 5 examines conditions on f,
for a Tychonoff topological space X.
X, and Y under which f,
Section 6 gives conditions under which
f
is continuous, and
preserves certain quotient-type mapping
453
STAR COMPACTIFICATION
"open", "proper",
properties, such as
and "perfect".
DECOMPOSITION SERIES.
2.
[7]
The reader is asked to refer to
for convergence space notation and
terminology not given here, as well as additional information about the star
[7],
As in
compactification.
space will always mean convergence space, and the
The separation axioms T
abbreviation "u.f." is often used for "ultrafilter".
I
(singletons are closed), T 2 (convergent filters have unique limits), and T 3
(regular plus T 2) will be used, but no separation axioms are assumed unless such
is explicitly stated.
X, let F(X) (resp. U(X)) be the set of all filters (resp.,
Given a space
X*
X U
X’.
}
3"
then
be the set of all non-convergent members of
If A c X, define
E F(X), and F’ #
{F’: F
X’
Let
ultrafilters) on X.
,
for all F
*
3’;
{
6
X’: A E,}, and A*
3"
otherwise,
A U A’.
If
be the filter generated by
then let
{F*
be the filter generated by
let
3
A’
(X), and
:F
3 }.
If
3’
exists,
We omit the easy proofs of the first
3
two propositions.
The following equalities hold for any subsets A, B of X:
PROPOSITION 2.1.
A’ U B’
A’
(AUB)’
B’
F(X*),
Let X be a space,
generated by {A
!
If
(c)
If
6
X,
X’
X*-,
-+
+
x
(A
Define
(a)
If
F(X),
(X)and
3’
exists, then
(X*)
X’
6
in
and
X*
,
6
then 6
iff there is
>_
iff
it is proved in
B*
X’
and
A convergence structure is defined on
For x
U
A’ E }.
X
PROPOSITION 2.2.
(b)
A*
(AB)’;
[II]
* <*
that
Let
i*
F(X*),
and
.’)^
3
6
X*
UB)*
^
A*
B* =(AB)*.
to be the filter on X
>_
*,
then
^ >-
>-
*;
for
(X).
as follows:
+ x in
X such that
denote the identity embedding of X into
(X*,i*)
is a compactification of X which is T 2
G.D. RICHARDSON and D.C. KENT
454
It is immediate from the construction that, for any non-
whenever X is T2.
X*-X
compact space X,
is a T
2
whenever X is pretopological.
pretopological space; thus
The universal property of
X*
is pretopological
K* established
in
[II]
will be obtained in Section 3 as a corollary of a much more general result.
A subset A of space X is bounded if each ultrafilter containing A is
X is said to be locally bounded if each convergent filter contains a
convergent.
X is essentiall7 bounded if 3 (
bounded set.
:
X’
(
and
% 3}
X’
implies that the filters
contain disjoint sets.
The next proposition is
[7].
proved in
PROPOSITION 2.3.
(b)
(a)
X is locally bounded iff X is open in
X is essentially bounded iff
X*-
X*.
X is discrete.
We shall next consider the relationship between the closure operators of X
and
,
cl
X*.
cl
X
be the closure operator on a space X.
For an ordinal number
we deflne:
x
A =A
I
clx A
cl
Let
X
cl A
X
clx(Cl-IA)
A
clA
U
<
c
A
if u-I exists
if
is a limit ordinal.
length of
between
the.
ED(X)
clA
u+IA
for all A c X is called the
cl
X
X
The relationship
decomposition series of X and denoted by
The smallest ordinal u such that
ED(X).
and
ED(X*)
can be obtained with the help of several lemmas.
For the remainder of this section, we shall assume that X is an arbitrary
space;
(X*,i*)
will always’ denote the star compactification of X.
smallest infinite ordinal number.
LEMMA 2.4.
n
If A c X, then c i
x,A
cl
n
XA
n-I
U (cl X A)’.
Let
be the
455
STAR COMPACTIFICATION
It suffices to prove this result for n--2.
PROOF.
CI2xA U (clXA)’ c__ cl2x, A
x
so
(X*)
such that
CI2xA.
If
:
(c].
2
B
F(X).
such that
0 X’
then it is easy to see that cl
cl, B
/
(clx* B)
>- 8’,
If
filter
+ x
X’,
_c X’,
(clx, B)
x
cl,B
,
clx, B.
{
XA
and
3
(X*)
Thus 8
(SF) F
(cl-2
U B’.
/
>- (6^)’.
>-
Then
{
B’ and x
>_
A similar argument shows that
{
X*
8 in
8 >-
Let
*
B~ c__ clx,
and B
B’
>
and so
and so
3
.
8F
B v.
containing B and a
(X) such that
{
(X)
{
B’)’.
we have
Letting 8
B
note
If
This implies
is an u.f.
B such that F
8F
such that
8
B ", then there is
choose
B
3
(B~)
and
BY) U
B};
{
X
By Proposition 2.2,
(X) and
8.
{x
B
Clx, B
3:
0
B
and B
(cl-I
B~ U
for all G
’
6
Also, 6
sections of the net
implies
B.
-
/
>_
x and
B"
be the filter of
x in
X*,
and so
This establishes the
I.
result for n
2
clx, B
If n--2, then
with the help of Lemma 2.4.
B"
B}
X, then there
>_
such that
for all G
For each F 6
(B~)"
then
let
Note that (B~)
}.
then there is
Conversely, if x
x
B
and so B O G’ #
X.
+ x in
B N G’ #
>_
X’
If B c
{ 8 X’: 8 >_
x in X and
If B
0
B
6
Thus
For this purpose, it is necessary to
For n--I, the statement becomes
PROOF.
8
Q implies
c__ X’.
for B
Define B
LEMMA 2.5.
Thus Q
QIx >_
and X
introduce some additional terminology.
8{
then there is
XA)’. B
We next describe
that
{(cl2x,A) 0 X
8"
>
clx, A
{
If x
is obvious.
Note that
B
c__
cl
xB
clx,
(B
U B v)
(B~)
U (B~)
By the remarks preceding Lemma 2.5, (B~)
This establishes the result for n--2.
tion to arbitrary n is now a trivial induction argument.
COROLLARY 2.6.
If
U (cl xBv)
A_c X,
then
clx, A
clx, A*
follows
B
and
The generaliza-
G.D. RICHARDSON and D.C. KENT
456
PROOF.
Clx, A*
clx, A
clx, A’. By Lemma 2.5, clx, A’
A’ _c clx, A, and (A’)
U
It is easy to check that (A’)
If A c X and u is a non-limit ordinal, then
LEMMA 2.7 (a)
PROOF.
Using transfinite induction along with Lemma 2.4, the results follow
7 +I, where
easily from both limit and non-limit ordinals, except for the case u
y
In this case we have
is a limit ordinal.
c’Y’+l
"X* A clx, (clYX A
U B)
cl
u
A)’: B <
X
c__ (c17XA)’; we omit
+I
A U
cl A)
where B
and B
The symbol
AX
U B"
U B
,:o
B"e
ho,,
the de tails.
represents the topological modification of X, i.e., the finest
topological space on X coarser than X.
THEOREM 2.8. AX is a subspace of
PROOF.
Let X
AX*Ix
I
clx, A clxA U A’
closed in AX* and A
A*0
Then
THEOREM 2.9.
(a)
(b)
ED(X)
>_
Let A
c__
If
PROOF.
Then
AU A’
%X*.
AX >_ X 1 is clear.
A*
clxA*
Let A
c__ X
be %X-closed.
by Corollary 2.6.
cl,
+ I.
is idempotent.
then
X*
and A
ED(X)
ED(X*)"
B U C, where B
ANX, C
AX’.
Then, by
clxk,+l
must be idempotent.
It follows easily from Lemma 2.4 that
clkx
ED(X*)
U C
is idempotent if
Thus (a) is established.
If X is a topological space, then
C
Thus
Statement (b) follows easily from Lemma 2.6.
COROLLARY 2.10.
is
ED(X*) _< ED(X)+ I.
is idempotent, it follows that
ED(X*) ED(X)
A*
X, which implies A is X 1 -closed.
If i _< ED(X) < a
then ED(X) _<
n-2
cl
Thus
-<
2.
If
STAR COMPACTIFICATION
We define a
for all
[4].
{
X’;
to be a regular space with the property that cl
this concept (but not the terminology) was introduced by Gazik
Discrete spaces and compact regular spaces are the most obvious examples of
G-spaces.
Another class of G-spaces are the ultraspaces, which are the topological
spaces having exactly one convergent ultrafilter.
[4]
-
X
457
in the case where X is
removal of the T
T2;
2
The next theorem is proved in
assumption causes no problems in
the proof.
X*
THEOREM 2.11.
is regular iff X is a G-space.
A regular space is symmetric if
of symmetric spaces include T
3
y and
x.
Examples
spaces, regular topological spaces, and regular
[14]
It is shown in
convergence groups.
- -
+ x whenever
that a compact symmetric space has the
same ultrafilter convergence as a compact regular topological space.
PROPOSITION 2.12.
If X is a symmetric G-space, then
PROOF.
and
-
Let
clude that both e and
*.
>_
8
+
8
are in X.
Since X is symmetric,
THEOREM 2.13.
(a)
in
X*
+
8
in
X*
(c)
If X is a T topological space,
2
If X is a G-space, then
there is
X, and so
/
/
D(X*)
8 in
_< 2 and
X*
we can con-
e in X such that
X*.
_< 2.
D(X)
D(X*) _< i.
then D(X*) _< I iff X is
If X is a symmetric G-space, then
(a)
,
If X is a G-space, then
(b)
PROOF.
is symmetric.
By the construction of
Since
/
X*
a G-space.
is a compact regular space by
Theorem 2.11, and it follows by Theorem 2.4(a),
[14],
that
D(X*)
2.
The
second inequality follows from the first and Theorem 2.9.
Statement (b) follows immediately from Proposition 2.12 and Theorem 2.4(b),
[4].
(c)
by (b).
If X is a topological G-space, then X is symmetric, and so
If
is regular.
D(X*)
_<
I, then X
is a compact T
By Theorem 2.11, X is a G-space.
D(X*)
and hence
2 topological space,
_< I
X*
G.D. RICHARDSON and D.C. KENT
458
%D(Y)
It should be noted that
%D(X*)
If X is not a finite discrete space,
0 iff X is a finite discrete space.
we can replace
3.
0 iff Y is discrete, and consequently
"D(X*)
_<
l"by
"D(X*)
I"
in (b) and (c) of Theorem 2.13.
R-SERIES.
In this section, we summarize some results on the R-serles of a space
[13]
(originally studied in
[14]),
and
X*,
obtain a few results on the R-serles of
and lay the groundwork for many of the results of Section 4.
Starting with a space X, an ordinal family of spaces
r0(X
same underlying set as follows:
-
rX iff
+ x in
The family
r
then
ryX
is denoted
R-serles of X and denoted by
[13]
shown in
that X
r
R(X).
Xr,
and
>_ cln
X such that
x in X and
X} is called the R-series of X.
rX r+ 1 X,
that
+ x in
-
N,
there exist n
is defined on the
X
N and
iff there exist n
rlX
x in
{r X}
<
X
cln_x
>_
such that
is the least ordinal such
If
is called the
length of the
R(X)
Note that X is regular iff
is the finest regular space coarser than
X, and Xr
0.
It is
is called
the regular modification of X.
even when X is T
X will not in general be T
2. A T 3 space
2
r
is constructed as follows.
associated with X (we use T to mean regular plus T
Of course
I)
3
First, define an equivalence relation among the elements of X by
in X
r
Let X
s
X
by
iff
/
be the set of equivalence classes with the quotient convergence
structure derived from X
If f
x
-
r
Y, let
Xs
+
following diagram coute:.
X
Y
----+
X
Y
Y
s
be the (unique) function which makes the
x
r
r
---+
X
--+
y
s
s
y
STAR COMPACTIFICATION
where the maps from X to X
X’ Y
and Y to Y
r
r
459
are the respective identity maps, and
are the respective quotient maps.
Let oX denote the symmetric modification of X, i.e., the finest symmetric
space coarser than X.
[13]
The next proposition follows immediately from results of
[14].
and
PROPOSITION 3. I.
If
X
f
-
Y is continuous, then each map in the following
commutative diagram (in which all non-labeled vertical maps are f and all non-
labeled horizontal maps are identities) is continuous.
X --+
R1 X
--+
y
y
rl
--+
--+
PROPOSITION 3.2.
PROOF.
X
--+
r( Y
--+
--+ r
--+
Let X
I
--+
--+
For any space X,
gX
X
--+
--+ gY
--+
Xr --+
Y
r
rlX
be the restriction of
I
,
in X
rlX*
On the other hand, let
I.
cl,
N such that
and n
in X such that
>_
*
by Lemma 2.4 and Corollary 2.6.
in
-+
>
Since
i*
:r X
I
x in X I.
,
ClX,n
rlX*.
i*
r
I
X --+
X*
X*
there is
cl ,
is
is continuous,
Q --+ x
Then there is
clx,n
>_
X*
--+
(cl-l’),
cl
and so
x
--+ x
rlX.
PROPOSITION 3.3.
of r
2
If X is a locally compact T
3
space, then r X is a subspace
2
X*.
PROOF.
It is sufficient to show that for any A
for all n 6 N, and’this will be proved by induction.
follows by Proposition 3.2.
then there is
F
-
--+ x in
F(X),
Since
Ys
to X
Since
and so we have
s
is a subspace of
continuous, it follows from Proposition 3.1 that
and thus r X >_ X
X
--+
c__ X,
clix (clix,
For n
A
I, the equality
n+l
If the equality holds for n and x
x and k { N such that
Since X is locally compact and
(clxk, F) (clIX, A)
T3,
A)N X
(Clr I X* A)
#
for all
we may assume without loss of
0 X,
G.D. RICHARDSON and D.C. KENT
460
From this observation, along with the induction hypothesis, we may conclude that
x
In+l
c+/-rl X A
It
is not true in general that
rX is
a subspace of
fact, we need to make use of some theorems from
rX*
To establish this
[9].
A space X is defined to be completely regu.lar if it is a subspace of a
symmetric compact space.
It is shown in
[9]
that X is completely regular iff it
is a symmetric space with the same ultrafilter convergence as a completely regular
topological space.
Let mX denote the finest completely regular space coarser than
A space X is defined to be 0-reular if
X
[9]
proved in
--+ x implies claX --+ x.
It is
that X is m-regular iff X is a subspace of a compact regular space.
The m-regular spaces include the completely regular spaces and also the c-embedded
spaces of Binz
[I].
PROPOSITION 3.4. (a) If X is a regular space which is not m-regular, then X
r
is not a subspace of
(b)
R(X*)
>
X*r
and
R(X*)
If X is a locally compact T
>_ 2.
3
space which is not m-regular, then
3.
The first part of (a) follows from the aforementioned character-
PROOF.
ization of m-regular spaces as subspaces of compact regular spaces.
statements concerning
R(X*)
The two
follow by Proposition 3.2 and 3.3, respectively.
For any space X, let C(X) be the set of all continuous real-valued functions
on X.
A T
3 topological space X for which C(X) consists only of constant functions
is an example of a regular space which is not m-regular; for this space,
and
R(X*)
>_ 2.
An example of a regular space X for which
R(X*)
R(X)=0
>_ 3 is obtained
with the help of the following lemma.
LEMMA 3.5.
points in X.
A locally compact T space X is m-regular iff C(X) separates
3
461
STAR COMPACTIFICATION
0X, then wX is T2 and so C(X) separates points in X.
If X
PROOF.
--+ x in
Conversely, if
contains a compact set A.
X, then
separates points in X, wX is T 2 and so the subspaces
It follows that cl
X
ultrafilter convergence.
Since
XIA and XIA have the same
Clx
--+
x in X, and therefore
X is w-regular.
EXAMPLE
Let X be the set
3.6.
[0, I]
If s is a sequence on the set X,
Let
denote the filter generated by s in the usual way.
let
().
generated by the sequence
(I) If x
{0, I},
then
-+
Define a convergence on X as follows:
x iff there is a sequence s converging to x in the
>_
topology such that
usual
(2)
--+
0 iff there is a sequence s con-
verging to 0 in the usual topology, but not a subsequence of
>-
s
(3)
>_
I iff
--+
the usual topology, or else
is locally compact and
T3,
be the filter
s
>_
(),
such that
where s is a sequence converging to 1 in
I.
One may easily verify that the space X
but C(X) will not separate the points 0 and i.
Thus,
by Lepta 3.5, X is not w-regular, and it follows from Proposition 3.4 that
R(X*)
>_ 3, whereas
Theorem 2.9.
R(X)
0; this result contrasts with the conclusion of
One can also show (we omit the details) that
r X =X, it follows that r X is not a subspace of r
3
3
3
X*.
6--+
I in r 3
X*.
Since
This shows that the con-
clusion of Proposition 3.3 cannot be improved without imposing additional
conditions.
[4]
Gazik showed in
topological space.
that a T
3
pretopologlcal G-space is a completely regular
Another result along these lines is
PROPOSITION 3.7. (a) A symmetric G-space is completely regular.
(b)
Every G-space’ is w-regular.
PROOF.
These statements follow immediately from Theorem 2.11, Proposition
2.12, and the characterization of w-regular spaces, obtained in
[9].
G.D. RICHARDSON and D. C. KENT
462
THEOREM 3.8.
Let X be a space.
X*
(a)
X is regular iff X is a subspace of r
(b)
X is e-regular iff X is a subspace of
(c)
X is completely regular iff X is a subspace of
I
X*r
X*.
Statement (a) follows immediately from Proposition 3.2.
Proof.
Statements
(b) and (c) are proved in [9].
For topological spaces X and Y, function f
X
--+
Y is defined to be
@-continuous if, for every x E X and every neighborhood V of f(x), there is a
PROPOSITION 3.9.
X
f
Then
--+
PROOF.
f
f
Cly
X
f
--+
Y, where X and Y are topological spaces.
X
--+
r
I X
--+
rlY
is continuous.
Y be @-continuous, and let
X such that
>-
cl
>-
x
cl
x b(x),
--+
where
x in
rlX.
Then there
(x) is the neighborhood
By @-continuity, f(cl b (x) >X
filter at x.
f() >_
Let
Y is @-continuous iff f
Let f
--+ x in
is
f(c %< c_ Cly V.
of f(x) such that
neighborhood
b(f(x)).
The latter filter
:rlX --+ rlY is continuous.
rlX --+ rlY implies f(clx
equivalent to 8-continuity of
cly b(f(x)), and so
rlY-converges to f(x), and
so
For the converse argument, one easily see that
(x))
f
X
>
Cly
--+
b(f(x)) for all x E X, which is
Y.
The characterization of 8-continuity given in Proposition 3.9 is not suitable
for a purely topological investigation, since
when X is topological.
f
X
if f
--+
rlY
rlX
--+
fail to be topological even
Perhaps this suggests that convergence spaces are the
natural realm for the study of 8-continuity.
a function
rlX may
But in any event, we shall define
Y between arbitrary convergence spaces to be @-continuous
is continuous.
More generally, if P is any function property, then f
to have property @-P if f
rlX
8-open maps, @-quotient maps, etc.
--+
rlY
has property
P.
X
--
Y is defined
Thus, one can speak of
Some of these "@-properties" will be studied
STAR COMPACTIFICATION
463
briefly in Section 6.
4.
NATURAL EXTENSIONS.
Let X and Y be spaces and consider a function f
X*
--+
Y*
A function
X --+ Y.
is called a natural extension of f if the following conditions are
satisfied:
(2)
X’
If
and f( is convergent in Y, then
f,()
is an element of Y to
which f () converges.
(3)
X’ and f()
If
X’
If
f
f,
X
--+
Y’
then
Y’,
implies f(9
Y’
f()
f,()
then f is said to be weakly proper...
Y is a weakly proper function, or if Y is
T2,
then the natural extension
is unique; in general, f may have many natural extenm, ions.
and theorem that
follow
when f
X
. - . Y,. f,
If
In the proposition
will be assumed to be an arbitrary
natural extension of f.
The proof of the next lemma is straightforward and will be omitted.
LEMMA 4.1.
If f
X --+ Y and A c X, then
(f(A))* c__ f,(A*)
(a)
If f is continuous, then
(b)
If f is continuous and Y is
T2,
(c)
If f is weakly proper, then
f,(A*)
THEOREM 4.2.
If f
then
c
c_i
f,(Clx, A) c_ cly,
f(A);
f,(clx, A) =cly, f(A);
(f(A))*.
X --+ Y is continuous, then
f,
X*
--+
Y*
is
0-continuous.
PROOF.
cl, f().
It is sufficient to show that, for each
,
(cl,
*
F(X), f,
U [cln-I F)’ by Lenna 2
x
cl,F el,
clF
and Corollary 2.6. By continuity of f, f,(cl F)
f(cl F) 5_ cl f(F). and
f,(cl-I F)* ely, f(cl-I F)c cly, (cl-If(F))c cln, f(F) follows with the
help of Lemma 4.1. Thus f,(cl, F*) c__
cln=** f(F), and the theorem is proved.
If
F (
then
F*
G.D. RICHARDSON and D.C. KENT
464
COROLLARY 4.3.
X
If f
Y is continuous, then each map in the following
--+
commutative diagram (in which all non-labeled vertical maps are
f,
and all non-
labeled horizontal maps are identities) is continuous.
r
r
I
X*
--+
-
1
Y*
--+
--+ r
r X*
--+
--+
X*r
--+
oX*
Y*
--+
--+
Y*r
--+
oY*
e
T<
X* s
Y*
--+
s
y*
The next resu.t closely resembles, but is more general than, the extension
[II].
property of the star compactification obtained in
COROLLARY 4.4.
then
X* --+ Y
f,
X
If f
--+
Y is continuous and Y is compact and regular,
If Y is also
is continuous.
Under the given assumptions, Y
PROOF
T2,
Y*
then the extension
r
I
Y*
since
X*
f,
>-
r
is unique.
I
X*
the
The second follows from an earlier remark.
first statement is establis’hed.
In the next section we shall see that continuity of f :X- Y does not
guarantee the continuity of
X
r
I
X*
X*
f,
--+
then by Proposition 3.2
<~
Our study of the compactification
Y*
<~
If X is a regular space, let
(X~, i*)
is a compactification of X.
will be limited to the following proposition.
PROPOSITION 4.5. Let X and Y be regular spaces.
(a)
If f
(b)
X
is T
2
is T
iff X is a T
If X is a T
2.
2
f,
X
--+
2
G-space, then
X*
on
X*
generated by
It is shown in
compactificatlon.
[12]
Th_-is
2.10, and so X
is regular by Proposition
X’ such
If X is not a G-space, then there is
X
]If
(X) and
x, then --+
#
1
is continuous.
G-space.
--+
filter
Y~
Statement (a) follows immediately from Theorem 4.2.
PROOF.
(b)
Y is continuous, then
X
converges in
in X
X*
~.
> cl
X
that
X’,
If
where
then the
and
to both
that every completely regular T 2 space has a
compactification is regular and
X*
T2,
Stone-ech
has the universal
property relative to the class of completely regular spaces, and agrees with the
465
STAR COMPACTIFICATION
COROLLARY 4.7.
X
If
is a completely regular T
space, then
2
(X%, i,-)
is
the Stone-Cech copactification of X.
If X is a Tychonoff topological space, then Corollary 4.7 gives a new method
8X.
for constructing
5.
Indeed, 8X is in this case the pretopological modification
CONTINUITY OF NATURAL EXTENSIONS.
We next consider conditions under which a natural extension f, of a continuous
For this purpose, we use some additional notation and
function f is continuous.
terminology.
Let f
X
extension of
--+
Y be a continuous function, and let
A’ f
For A c X, define
f
A’
{
f,(A’f)
PROPOSITION 5.1.
note that
f,
is continuous at x
(2)
f,
is continuous at
(3)
f,
is continuous at
If 3
Y*
be a natural
Y.
F(X) define
If
F
Let
};
F(X) is said
Let f be a continuous map.
(I)
PROOF.
--+
f) converges in Y}.
Ff,(A) f,(A*)
6 F(Y) to be the filter generated by
{Ff,(F)
to be f,-closed if
Ff,() f(3).
f(A) U
Ff,(A)
Ff,()
X*
f,
f-l(y)
iff
-+
x in X implies that
-I (Y) N X iff
--+ f,
I
is f,-closed.
f, (g,) iff
(
F(X) then one can easily show that
Ff , 3
--+
f(x) in Y.
in Y.
rf,()
f,
Ff,
>_
(Ff
))
(I) and
(2) follow from these inequalities.
(3)
If
and hence
thus
f,
Ff,
f,
is continuous at
5= f().
Conversely,
f,
is continuous at
COROLLARY 5.2.
Let f
X
i
--+
f,-I (y,),then Ff, 3 >_ f,(3*) >_ (f(3))*,
,
f,(3*) >- (Ff)* (f)* f, in Y and
--+
(y.).
Y be continuous, and Y a regular space.
(I)
f,
is continuous at all points of
(2)
f,
is continuous iff each
-i
f,
(Y).
-I
f, (Y’)
is
f,-closed.
Then
466
G.D. RICHARDSON and D.C. KENT
Stone-ech
topological
X is topological.
compactification relative to ultrafilter convergence when
We shall now give an alternate construction of this compactifi-
,
K
cation using
X%
For any space X,
generally true that X
is a compact, regular, T
2
is a subspace of
X*_
space.
However it is not
Recall the notation oX
for the
symmetric modification of X.
THEOREM 4.6.
completely regular,
ech
Xs
is completely regular and T2, and
is the Stone-
i,
By assumption, OX is a subspace of a compact symmetric space, and
Xs
hence completely regular.
is T
i,
(X
X
the maps
--+
(X%
then gX is
compactification of X
s
PROOF.
that
oX*,
If X is a space such that oX is a subspace of
@X
and
@X*
if --+ e in
Xs
x in gX and
gX
--+
s
--+
2
by construction.
,
Xs
are strongly open (see Proposition 2.2,
and x
-i
X
X()
defined by x.
Corollary 4.3
extension of
--+
on X such that
s
is densely embedded in X s-
f([x]),
[x]
It is easy to check that F
X
X*S
--+
Y *S
Xs
--+
Y is continuous, then
is the equivalence class in X
--+
Stone-ech
Y is continuous
(F,)
is clearly an
compactification established in
that this compactification is equivalent to
s
Y is continuous, and so by
f, and this. extension is unique because Y is Hausdorff.
the uniqueness of the
oX*
,
where
Y by F(x)
(F,)
This means
Using this property and the fact that oX and
If Y is a regular, compact, T space and f
2
X
[14]).
(e)’ then there is a filter
are symmetric, one can easily show that X
define F
In the diagram that follows
(X%, i,-).
[12],
Thus by
it follows
STAR COMPACTIFICATION
(3)
If X is essentially bounded, then
f,
467
is continuous.
Statements (I) and (2) follow immediately from Proposition 5.1 and
PROOF.
The assumption that X is essentially bounded (see
the fact that Y is regular.
f,-I (y,)
Section 2 for this definition) guarantees that each
PROPOSITION 5.3.
X
If f
--+
Y is continuous and weakly proper, then f,
.
is continuous.
PROOF.
If f is weakly proper and A
F(X) is f,-closed.
nuit#
of
f,-closed.
is
c__
X, then
A’f
Thus each filter
The conditions (I) and (3) of Proposition 5.1 for conti-
are thus satisfied, while condition (2) is satisfied vacuously.
f,
COROLLARY 5.4.
embedding, then
PROOF.
If X is a closed subspace of Y and f :X
X
f,
--+
Y
--+
Y is the identity
f,
is continuous by
is also an embedding.
Since X is closed in Y, f is weakly proper; thus
Proposition 5.3.
F(X).
for all
is clearly one-to-one, and by Lemma 4.1,
f
From this equality, it follows easily that
f()*
f,(5*)
f,
is an
embedding.
PROPOSITION 5.5.
The following statements about a regular space Y are
equivalent.
(a) Y is a G-space.
(b) Every natural extension of every continuous function into Y is continuous.
(c)
If Z and Y have the same set, Z is discrete, and f
identity, then
f,
(a)
PROOF.
is continuous.
(b).
If f
X
is regular by Theorem 2.11 and so
--+
f,
Y is continuous and Y is a G-space, then Y
X*
--+
Y * is continuous by Corollary 4.4.
(b)
(c).
Obvious.
(c)
(a).
If Y is not a G-space, then there is
Since Y is T
extension
f,
1
Y is the
Z
and Z is discrete, it follows that F
Thus
is not
f,-closed,
f
Y’
#
such that
cly
for some natural
and by Proposition 5.1
f,
is not
#
G.D. RICHARDSON and D.C. KENT
468
continuous.
Te final result in this section
is analogous to Proposition
involves e-continuity rather than continuity.
5.1, but
Since this result is of marginal
interest, we shall omit the proof.
PROPOSITION 5.6.
Let f
X--+ Y be a e-continuous map.
(a)
f,
is e-continuous at each point x
X.
)
f,
is e-continuous at
_- (y)
f,l
X’
f,
is e-continuous at
f
6.
n
--+
X
f,
in
fly
for
I.
each n
(c)
iff f(cl
>_
x
1
f,
(y,) iff, for each n
I, there
1 such
is m
).
f
QUOTIENT EXTENSIONS.
In this concluding section, we shall consider the circumstances under which
f, will possess
certain quotient-type properties.
We begin with definitions of
the properties to be considered.
The term map will be used to mean a continuous, onto function.
f
X
I.
-
f is proper if
on X such that
2.
X*
Y is onto, then any natural extension f,
f()
,
--+
f is a map and, whenever
then there is
x
f-l(y)
--+
Y*
y in Y, and
f-l(y)
f is perfect if f is a proper convergence quotient map.
4.
f is open if f is a map and whenever
5.
f-l(y),
then there is an u.f.
f is closure-preserving if A
!
-
is an u.f. on Y which
P
is said to have property
y in Y,
converges
x such that f()
X implies f(cl A)
X
cly
Further information about these properties may be found in
Recall that if
--+
f()
--+ x in X such that
and
is an u.f.
--+ x.
such that
3.
to y, and x
[10].
is also onto.
f is a convergenc e quotient map if f is a map and, whenever
there is x
Note that if
f(A).
[6], [8],
and
represents any of the above properties, then f :X
e-P if
f
r
iX
--+ r
lY has
property
P.
--+
Y
469
STAR COMPACTIFICATION
It is clear that a proper map is weakly proper, and that a weakly proper
map onto a T 2 space is proper.
In all of the propositions that follow,
X into a space Y, and
PROPOSITION 6.1.
PROOF.
X*
f,
If
is an u.f. on X
Suppose that
Suppose that G---
t
X*
>_
--+
in
and
proper map, and thus
.
Hence
and
N*
Y, then it follows that
If
f,()
N*,
>_
f()
>_
.
*
a In
on X such that
f()*
f(*)
Y*
c in
where
then there exists a f.lter
*.
f(.)
is also a proper map.
since
f is a
This implies
are not disjoint filters on
are not disjoint filters on Y; consequently, f()
that f() and
map,
X*
f,
such that
f()
on Y such that
in
f.
denotes a natural extension of
f is a proper map, then
Then there exists an u.f.
Y*
y*
--+
f denotes a function from a space
is a proper map.
PROPOSITION 6.2.
If
f
_
f-l(c),
x in X for some x E
.
f,()
E Y, then since f is a proper
If
---+
and thus
E.
x in
X*.
It follows that f,
convergence quotient map, then f, is a conver-
is a
gence quotient map iff f, is continuous.
PROOF.
hence
The relation
<_ f(
f,(
and
Y; let
f,(*)
_<
*.
--+ x in
f,(
<_
)*
f(A)*
is satisfied for each subset A of
on X.
for each filter
be any u.f. on X such that f()
If
E Y, then there exists
X and f()
*,
f,(A*)
since
it follows that
f,
Suppose that
=
Then
E F(X) and x E
*
f,()
f-l()
f is a convergence quotient map.
is a
--+
X, and
in
and
such that
Since
convergence quotient map precisely when f,
is a continuous map.
COROLLARY 6.3.
PROPOSITION 6.4.
0-open, and 0-proper.
f,
Y*
is a perfect map whenever f is a perfect map.
If f is an open, proper map, then
f,
is open, proper,
G.D. RICHARDSON and D.C. KENT
470
It follows by Proposition 6.1 that f, is proper.
PROOF.
[14],
then it follows from Theorem 4.2,
f,
it remains only to show that
Let
(Y*)
6
>_
*
--+
and
hence there exists exactly one u.f.
,
f()
Y*,
in
*
3"
1
f, ()
and
o-proper.
is also o-open and
then
and let
--+
in
1
f, ().
8
Y*.
Thus
3
f,( 3*)
8
* -<
X*.
in
Let
6
and 8 --+
f,(8)
Then there is
Y, then 8
If
on X such that 3--+
and since f is a proper map,
contain both
f,
is also open,
f,
is open.
such that
(Y) such that
that
If
8
X, and
Thus
(X*)
8
X*.
in
Y then, since f is a proper map, 8 6 X, and since f is an open map,
If e
--+
?(X) such that
there is
8
in X and f(3)
Then, as in the argument
8
of the preceding paragraph, there exists an u.f. 8 --+
this establishes that
in
X*
such that
f,(8)
is an open map.
f,
We omit the straightforward proof the next proposition.
PROPOSITION 6.5.
If f is perfect 8-proper map, then
f,
is a 8-perfect map.
Propositions 6.4 and 6.5 yield the following corollary.
COROLLARY 6.6.
If f is an open perfect map, then
f,
PROOF.
--+
is a 8-convergence quotient map.
Suppose that
e and
>_
cln,
Y
.
,
--+
in
F(Y) such that
Then there is
rlY
Y’.
Y, then it may be assumed that e--
If
(X) such that f()
Let 3 6
is a 8-perfect map.
If f is a convergent quotient map which is closure preserv-
PROPOSITION 6.7.
ing, then
f,
then 8
3
X’
.
f,(
and
By Lemma 2.4
,
and the assumption that f is closure-preserving, it follows that
f*(cln:Ix 3*) -< f,((cl
.
3)*)
Cly, f(cl
nx
3)
then, since f is a covergence quotient
that
in
f()
rlX*.
Thus
f,
If
is a
I
)
f,(clnl
X
f,-l() v cln,+l
X
Again,
-convergence
_<
n
Cly, Cly f(,)
map, there is --+
cln, *.
Y
quotient map.
then
--+
In both cases
8
in
rlX*
and
cl
8
,
f
1
cln:
X-
If
-I () such
*
f,(E)
STAR COMPACTIFICATION
The final proposition follows immediately
PROPOSITION 6.8.
471
fom
Proposition 5.6.
If f is @-continuous and closure preserving, then
f,
is
@-continuous.
We conclude by citing, without detail, some examples which place limitations
on the types of results obtained in this section.
Example 4.3 of
[14]
that in this case
The function f constructed in
is perfect but not @-proper; it is also not difficult to show
f,
is not
assumption that f is open.
@-proper.
Thus, in Corollary 6.6, one cannot drop the
There are other examples which show that
f,
to be continuous when f is an open, convergence quotient map, and that
to be open when f is open and
f,
may fail
f,
may fail
is continuous.
REFERENCES
[I]
Binz, E., "Continuous Convergence in C(X)," Lecture Notes in Mathematics
Berlin, Heidelberg, New York: Springer-Verlag, Vol. 469 (1975).
[2]
Dickman, R.F. and J.R. Porter. "8-Perfect and 8-absolutely Closed Functions,"
Illinois, J. Math 21 (1977) 42-60.
[3]
Fomin, S., "Extensions of Topological
[4]
Gazik, R.J., "Regularity of Richardson’s Compactification," Can. J. Math.
26 (1974) 1289-1293.
[5] Herrmann, R.A., "Perfect
Soc.
Quotient
Maps,"
Fund. Math. 65 (1969) 197-205.
D.C. and G.D. Richardson, "Compactifications of Convergence Spaces,"
Internat. J. Math. and Math. Sci. 2 (1979) 345-368.
[8]
’Open
Math
[9]
Maps on Convergence Spaces," Bull. Austral. Math.
(To appear).
[6] Kent, D.C., "Convergence
[7] Kent,
Spaces," Ann. Math. 44 (1943) 471-480.
and Proper Maps Between Convergence Spaces,
Czech
23 (1973) 15-23.
"Completely Regular and e-Regular Spaces," Proc
Conv. Structures, Cameron Univ., Lawton, OK., April
Conference on
84-90.
190,
[i0] Lowen-Colebunders, Eva, "Open
Setvalued
Maps," Can. J.
and Proper Maps Characterized by Continuous
Math. (To appear).
472
G.D. Rr CHANDSON and D. C. KENT
[11] Rchardson, G.D. "A Stone-ech
Copactification for Limit Spaces " Proc
Aer. Mth. Soc. 25 (1970) 403-04.
[12]
Richardson, G.D. and D.C. Kent, "Regular Compactiflcations of Convergence
Spaces," Proc. Amer. Math. Soc. 31 (1972) 571-573.
"The Regularity Series of a Convergence
Math. Soc. 13 (1975) 21-44.
[14]
Space,"
"The Regularity Series of a Convergence Space
Math. Soc. 15(1976) 223-244.
Bull. Austral.
II,"
Bull. Austral.