C OMPOSITIO M ATHEMATICA
D AVID K. H SIEH
A characterization of the B-realcompact space
Compositio Mathematica, tome 29, n
o1 (1974), p. 63-66
<http://www.numdam.org/item?id=CM_1974__29_1_63_0>
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Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques
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63
A CHARACTERIZATION OF THE B-REALCOMPACT SPACE David K. Hsieh
Noordhoff International Publishing
Printed in the Netherlands
1. Introduction
Realcompact
spaces are most often studied and characterized in the framework of thering C(X)
of all continuousfunctions,
both bounded andunbounded,
defined on the space X. We abandon this traditional ap-proach
torealcompact
spaces in favor of a moregeneral’yet simpler setting
made availableby
theÂ-compactification theory.
Thus instead of thering
of all continuousfunctions,
we consider a Banachalgebra
offunctions;
instead of both bounded and unboundedfunctions,
we needonly
to work with the bounded functions. Frolik introduced in[1]
the concept of acomplete family
for unbounded continuous functions andproperties
ofQ-spaces (i.e. realcompact spaces)
were derived in terms ofcomplete
families. Since we deal withonly
the bounded functions in the03BB-compactification theory,
Frolik’s definition ofcomplete family
cannotbe
adopted
here. In thisarticle,
we introduce withoutusing
unboundedfunctions the definition of
B-complete family
of bounded functions in terms ofpositive singular
functions in agiven
Banachalgebra
B, and thencharacterize the
B-realcompact
space, which is ageneralization
of therealcompact
space, in terms of theB-complete
families.2. Preliminaries
In
general
weadopt
the notation and definitions in[3]
and[4].
Thuswe
study
anarbitrary
set E and a Banachalgebra
B of bounded real- valued functions defined on E. The norm in B is definedby ~f~ = supx~E If(x)1
for eachf
in B. B is called an admissible Banachalgebra
if Bcontains constants and B separates
points
in E. Apositive
cone in B is acollection of
positive singular
functions in B which is closed under addi-tion,
andmultiplication by positive
scalars. A maximalpositive
cone(m.p.c.)
is apositive
cone notproperly
contained in any otherpositive
cone. A m.p.c. M is said to be
strong
if there exists a functionf
in Msuch that
f(x) ~
0 for each x ~E,
otherwise M is said to be weak. Am.p.c. M is said to be free if there exists no
point
x in E such thatf (x)
= 0for every
f
in M.64
3.
B-realcompact
spaces andB-complete family
of functionsIn the
following discussions,
E willalways
denote a set and B an ad-missible Banach
algebra
on E.3.1 DEFINITION: Let A be a subset of E and
f
be anon-negative
functionin
B. f
is said to bestrongly
bounded on Aprovided
that either there is apoint x
in A such thatf(x)
= 0 or there exists an e > 0 such thatf(x) ~
8for each x in A.
NOTATION : When
f
isstrongly
bounded onA,
we writef
0 on A.3.2 DEFINITION : Let 3° be a filter of zero sets of functions in B. ff is said to be a maximal s-filter in
(E, B) provided
that the setF- = {f
E B:f ~ 0, [ f e]
~ F for each e >01
is a m.p.c.(maximal positive cone)
in Bwhere
[ f, 8]
denotes the set{x ~ E: |f(x)| ~ el.
For latter
discussion,
we shall need thefollowing
lemma which isproved
in[4].
3.3 LEMMA : F is a maximal
e-filter if
andonly if given
anon-negative function f
in Bif for
every e > 0 the set[ f, 8J = {x ~ E: |f(x)| ~ 03B5}
meetsevery member
of
F then[ f, 03B5]
~ Ffor
every 8 > 0.3.4 LEMMA: Let M be a
m.p.c. in
B. LetM* = {[f, s]:
8 >0, f ~ M}
where
[ f, 8]
denotes the set{x ~ E : |f(x)1 ~ 03B5}.
Then M* is a maximal8-filter.
PROOF : Since M is a
positive
cone,by
definition[ f, 03B5] ~ ~
for eachf E M.
Now let[ f e]
and[g, 03B4]
be in M*.Clearly [ f, 8J
n[g, 03B4] ~ [f+g,
9 A03B4].
Finally
we show thefollowing:
ifZ(h) :D [ f 8J
for some[ f, 8J
in M*then
Z(h) E M*.
First we define a sequence gn in B asbelow;
for eachposi-
tive
integer n,
letIt can be
readily
verified that gn convergesuifcrmly
to a functiong(x) where g(x)
= 1 if x E[ f, aJ and g(x) = Ih(x)1
+elf (x) if x ~ [ f, a].Since B
is aBanach
algebra
with the norm definedby ~f~
=sup {|f(x)1 : x ~ E}, g
isin B. In
fact g
is anon-negative
function in B. We now claimgf is
in M.This claim follows from the observation:
(i) gf ~ ~g~f; (ii) ~g~f
E M; and(iii)
M is a m.p.c. HoweverZ(h)
isprecisely
the set[gf, a].
HenceZ(h)
E M.This
completes
theproof that
M* is a filter of zero sets. Since M is a m.p.c., M* is a maximal e-filterby
definition.3.5 DEFINITION : A
subfamily
ofnon-negative
functions H c B is saidto be
complete provided
thatgiven
a maximal e-filterF,
if for everyf E H
there exists aZf ~ F
such thatf
0 onZ f
thenn 57 + p.
3.6 THEOREM:
Suppose
F is a maximale-filter,
and suppose thatfor
each sequence
{Fn}
inF, ~~n=
1F n =1= 0. Then for
eachnon-negative function f in
B there is aZ f
in F such thatf
0 onZ f .
PROOF: For each
positive integer
n, let[f, 1/n]
denote the set{x ~ E:
|f(x)| ~ llnl.
We consider two cases.Case I :
Suppose
there is aninteger
k such that[ f, 1/k] ~ F. By
Lemma3.3,
there is someinteger t
and there is aZf ~ F
such that[ f 1/t]
nZ f
0.
Hencef
>llt
onZ f .
Case II: On the other
hand,
suppose[ f 1/n] ~ F
for each n. Sinceeach sequence in F has non-empty
intersection,
there is apoint
x in~~n=
1[ f; 1/n]. Clearly f(x)
= 0. Therefore in either casef
0 on somemember of F.
3.7 DEFINITION: E is said to be
B-realcompact provided
that every free m.p.s. in B is strong.REMARK : It can be
readily
verified that atopological
space X is real- compact if andonly
if X isC*(X)-realcompact
whereC*(X)
is the Banachalgebra
of all bounded real-valued continuous functions.3.8 THEOREM: The
following
statements areequivalent:
(i)
E isB-realcompact.
(ii) If!F is a
maximale-filter
in(E, B)
such that each sequence in F has non-empty intersection, thenn 57 + ~.
(iii)
The setB+ = {f ~ B: f ~ 01 is a B-complete family.
(iv)
G= {f ~ B : f t+0
onEl is B-complete,
wheref ~ 0
means thatf
> 0 andfor
each e > 0 there exists x ~ E such thatf(x)
B.(v)
Thereexists a B-complete family
H in B.PROOF :
(i) ~ (ii). Suppose
there exists a maximal a-filter F such that each sequence in F has non-emptyintersection,
but~ F = ~. By
definition3.2,
theset F- = {f
E B :f ~ 0, [ f, e]
~ F for each e >01
is a m.p.c. in B.Since
n 57
=9,
F- is a free m.p.c. For eachf ~ F-, [ f, 1/n]
~ F forevery
integer
n. Thusfl §J/=
1[f, 1/n] ~ 0. Suppose
that y is in~~n=1 [f, 1/n].
Clearly f(y)
= 0. That is !F- is a weak free m.p.c. in B. Hence E is notB-realcompact.
(ii) ~ (i). Suppose
E is notB-realcompact.
There exists a weak free m.p.c. M in B. LetM* = {[f, B]: B
>0, f ~ M}
where[ f, BJ
denotes the66
set
{x ~ E: |f(x)| ~ el.
It follows from Lemma 3.4 that M* is a maximal e-filter.Let {[fn, 03B5n]}
be a sequence in M*. We mayassume IIfnll
= 1for each n. Hence
f
=03A3~n= 1 fn/2n
is in M. Since M is a weak m.p.c., there exists x in E such thatf(x)
= 0.Hence fn(x)
= 0 for each n. It follows that x ~n:=
1[fn, 03B5n] ~ ~. But n M* = $J as
M is a free m.p.c.(iii)
~(ii).
It followsdirectly
from Theorem 3.6 and Definition 3.5.(ii)
~(iii).
Let F be a maximal e-filter.Suppose
for eachf
EB+,
thereexists
Z f
in J’ such thatf
0 onZf.
To show that B+ isB-complete,
it
suffices,
in view of(ii),
to show each sequence in 3F has non-empty inter- section. Assume the contrary: suppose that there exists a sequence{Zn}
inF such
that ~~n=1 Zn = 9.
Since 3F is a maximale-filter, F- = {f ~ B : f ~ 0, [ f, e]
~ F for every e >01
is a m.p.c. in B. WriteZn = Z(fn),
thezero set of
fn, with fn ~ B+
and~fn~ ~
1.Obviously [ fn, e] c- 5’
for each03B5 > 0 and each n. Thus
fn ~ F-
for each n.Define f = 03A3~n=1 fn/2n.
Clearly f
is in F- as 57 - is a m.p.c. But~~n=
1Z(fn) = ~,
it follows thatf
> 0. Now recall thatZ f ~ F,
and let e > 0 begiven.
Choose n solarge
that
1/2n
E. Since F is afilter,
there exists x inZ1
~ ··· nZn
nZf.
Clearly f(x) ~ 1/2n
03B5. This contradictsthat f 0
onZf.
(iii)
~(iv). Suppose J’
is a maximal F-filter and that for eachf
E G thereexists a
Zf ~ F with f 0
onZ f .
To show~F ~ ~,
letg ~ B+ ~
G.Being
the zero set of the zerofunction,
E is in F.Clearly g 0
on E whichis in F. Since B+ is
B-complete nJZ7 -7É 9.
(iv)
~(v).
Trivial.(v)
~(ii).
It followsimmediately
from Theorem 3.6 and Definition 3.5.REFERENCES
[1] Z. FROLIK: Applications of complete families of continuous functions to the theory of Q-spaces. Czech. Math. J. 11 (1961) 115-132.
[2] L. GILLMAN and M. JERISON: Rinfs of Continuous Functions. Van Nostrand, Princeton, 1960.
[3] D. HSIEH: Topological properties associated with the 03BB-map. Bull, de l’Acad. Pol. des Sc. XX (1972) 1-6.
[4] D.HSIEH : The 03B5-filters and maximal positive cones (to appear).
[5] E. LORCH: Compactification, Baire functions, and Daniell integration. Acta Sc. Math.
(Szeged) XXIV (1963) 204-218.
[6] J. MACK: Countable paracompactness and weak normality properties. Trans. Amer.
Math. Soc. 148 (1970) 365-272.
(Oblatum 8-XI-1973 & 13-V-1974) Mathematics Department City University of New York
The City College
New York, N.Y. 10031 U.S.A.