18
Lattice Measures and Topologies (*). RONALD CoI~,~ (Brooklyn, lq. Y., U.S.A.) Summary. - We abstract Frink's notion o] a normal base o/a topological space to an arbitrary lattice, and replace the ~wtion o] filters on a base by zero-one measures on a lattice. TMs oilers analytical simpli]ieation and elari/ieation, and extends to arbitrary measures as well. By putting a topology on the set o/measures, we generalize the notion o] Wallman.type eompac- tifieations, and we look at relations between the compaetifications by examining the u, nderlying lattices. O. - Introduction. It is well known (el. [20]) that there exists on a Boolean algebra a one-to-one correspondence between zero-one measures, ultrafilters, and maximal prime ideals (note: we shall adhere to the lattice terminology used mainly by SiKORSKI in [20] and SAMVE~ in [19]). This correspondence collapses when one deals simply with a lattice. This is adjusted by the introduction of regular lattice measures. In the first section we will state and prove the fundamental relationships between lattice ultrafilters and zero-one regular lattice measures, and lattice prime filters and zero-one lattice meas~tres. We study this interrelation and then carry out the development in terms of lattice measures. This offers an analytic simplification and clarification and enables us to get fairly complicated filter statements as simple corollaries. This is done in a number of cases of interest. More importantly, the resnlts can be applied to an atomic situation with appropriate lattices to obtain as corollaries important topo- logical results due to FROLIK [11], SULTAN [22], DYKES [8], BACH~A~ [4], :P]~TmlS [18], and others. We apply these results to numerous special eases, but is it clear that the examples could be greatly extended. We shall adhere throughout to the following terminology and notations: 53 will denote a complete Boolean algebra containing all l~ttices 1ruder consideration; ~ will denote a sub-lattice. A b-lattice is one closed under countable meets, a(C) is the algebra generated by ~; ff~(C) is the ring generated by C. The minimal element of :5 is denoted by o and the maximal element by e. For x e :5, x' is the (unique) complement of x in :5. A measure/z on a Boolean algebra is a non-negative real-valued function on A such that #(aVb)~ #(a)~-ju(b) for all (*) Entrata in Red~zione il 20 gennaio 1975.

Lattice measures and topologies

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Lattice Measures and Topologies (*).

RONALD CoI~,~ (Brooklyn, lq. Y., U.S.A.)

S u m m a r y . - We abstract Frink's notion o] a normal base o / a topological space to an arbitrary lattice, and replace the ~wtion o] filters on a base by zero-one measures on a lattice. TMs oilers analytical simpli]ieation and elari/ieation, and extends to arbitrary measures as well. By putting a topology on the set o/measures, we generalize the notion o] Wallman.type eompac- tifieations, and we look at relations between the compaetifications by examining the u, nderlying lattices.

O. - I n t r o d u c t i o n .

I t is well known (el. [20]) t ha t there exists on a Boolean algebra a one-to-one correspondence between zero-one measures, ultrafilters, and maximal pr ime ideals (note: we shall adhere to the lat t ice terminology used mainly b y SiKORSKI in [20] and SAMVE~ in [19]). This correspondence collapses when one deals s imply with a lattice. This is adjus ted by the in t roduct ion of regular lat t ice measures. I n the first section we will s ta te and prove the fundamenta l relationships between lat t ice ultrafil ters and zero-one regular lat t ice measures, and lat t ice pr ime filters and zero-one lat t ice meas~tres. We s tudy this interrelat ion and then car ry out the development in te rms of lat t ice measures. This offers an analyt ic simplification and clarification and enables us to get fair ly complicated filter s ta tements as simple corollaries. This is done in a number of cases of interest . More impor tan t ly , the resnlts can be applied to an atomic s i tuat ion with appropr ia te lattices to obtain as corollaries impor t an t topo- logical results due to FROLIK [11], SULTAN [22], DYKES [8], BACH~A~ [4], :P]~TmlS [18], and others . We apply these results to numerous special eases, bu t is it clear t h a t the examples could be great ly extended.

We shall adhere th roughout to the following terminology and notat ions: 53 will denote a complete Boolean algebra containing all l~ttices 1ruder considerat ion; ~ will denote a sub-lattice. A b-lattice is one closed under countable meets, a(C) is the algebra genera ted by ~; ff~(C) is the ring generated by C.

The minimal e lement of :5 is denoted b y o and the maximal e lement by e. For x e :5, x ' is the (unique) complement of x in :5. A measure /z on a Boolean algebra is a non-negat ive real-valued funct ion on A such t h a t # ( a V b ) ~ #(a)~-ju(b) for all

(*) Entrata in Red~zione il 20 gennaio 1975.

148 I~O~A~D COHEn: Lattice measures and topologies

a, b~ A with aAb-= o. I f a~$o implies t h a t lim#(a~) = 0, ~ is called ~-smooth. On a ring, e-smooth is equivalent to countable addi t iv i ty . # defined on a(E) (or any Boolean algebra or ring containing C) is called C-regular (just regular if there is no ambiguity) if # ( x ) = s u p { # ( a ) : a < x , a e £ ) for any x belonging to the par t icular Boolean algebra or ring containing C. Final ly we assume t h a t all lattices under con- sideration contain o.

1. - E-u]traf i l ters a n d [: -regular z e r o . o n e m e a s u r e s .

Most of the results of this section have already been given in :BACI-IMAN and CoHE~ [5], so we can afford to be brief in thei r development , as well as omit several proofs. Ze t F be an £-ultrafilter and let A ( F ) = ( x e S ~ : x > a for some a e F or x '>b for some be/V}. A(/~) is well defined, for if x > a E F and x '>be_F, t hen

o = max' > a Ab e ~, ~ contradict ion.

I J ~ A 1.1. - A(F) is a Boolean algebra.

Z ] ~ A 1.2. - A ( F ) o C (whence A ( F ) o a(C)) if and only i f / ~ is an C-ultrafilter.

P~ooF. - ~ e t F be an C-ultrafilter. I f b e C and if b ~ a ' for any a e ~ , t hen bAaV=O for any ae /V. Hence the fami ly {bAa: ae /~} is ~ filter base which gene- rates an ~-filter G o / v Thus b e G = / v as F is maximal .

The converse is a direct application of corollary 1.1. ~ o w on A(F) define

I 1 if x > a for some a e / ~ , #~(x)

l 0 if x ' > b for some b e F .

Then it is quite simple to show t h a t #F is an C-regular zero-one measure on A(F) .

LEM~A 1.3. - ~e t # be a zero-one E-regular me~sure on a(~), and let F = {ae C: #(a) = 1}; t hen F is an ~-ultrafilter.

Proof. - To show tha t F is not empty, suppose t h a t there exists some b e C such t ha t b ~ F . Then # ( b ) = 0, and so # ( b ' ) = 1; by regular i ty there exists a e £ with # ( a ) = 1, and therefore a e F .

Clear ly /v is a filter. Suppose there exists an C-filter G ~ F properly. Then there exists c e G such t ha t c ~ F . Then # ( e ) = 0, so t h a t # ( c ' ) = 1, and by regular i ty the re exists d e c such t h a t d<v' and # ( d ) = l . Therefore d e F c G , and hence sad = o e G~ which is impossible. Therefore /v = G. Summarizing, we have

T~EO~E~ 1.1. - There exists a one-to-one correspondence between all zero-one C-regular measures on a(C) and all C-ultrafilters.

~[~OI~ALD COHEN: Lattice measures and topologies 149

COI~O~LAI¢¥ 1.1. - L e t / 7 be an ~-filter; t h e n / ~ is an £-ultrafi l ter if and only if for each a e £ e i ther a e 17 or the re exis ts b e /~ such t h a t b < a ' .

This corol lary generalizes a resul t of ALb and SRAPII~0 [1].

DEFII~ITIO~ 1.1. -- I f to each a t o m b e ~ the re exists y e £ such t h a t b <y, t hen is called a t o m relat ing.

DEFINITION 1.2. -- We say t h a t ~ is a t o m dis junct ive if whenever we are given

a e ~ and an a t o m b e ~ dis joint f r o m a, t he re exis ts c e ~ such t h a t aAc-~ 0 a n d

b<e. Observe t h a t a t o m dis junct ive implies a t o m relating, a l though we shall usual ly

s t a te bo th for emphasis .

DEFII~ITION 1.3. -- We say t h a t an S-filter F is p r ime if given a, b e £~ t hen aVb ~ ~ if and only if e i ther a e / 7 or b e F .

THEOICE~ 1.2. -- I f ~ is a t o m rela t ing and a t o m disjunctive, t hen for all a toms

b e ~ , /Tb = (x e £: b < x ) is an ~-ultrafilter. I f ~ is not a t o m disjunet iv% then F b need only be a p r ime S-filter.

The following is ~n example of a non-dis junct ive la t t ice where/Tb is not an u l t ra- fi l ter: le t X be a n y infinite set wi th the cofinite topology: let :5 ~- if(X) and £ = the

open sets. ~ is not a t o m dis junct ive ( there are no disjoint open sets). Le t x e X ; /7~ is not an ~-ultrafilter. Fo r ~(x} is ~n open set which does no t belong t o / ~ and

whose complemen t does not conta in an open set in F~.

TtIE01~E~ 1.3. -- An ~-ultrafil ter has the c .m.p. (countable meet property, i .e. , if o o

a i e F , i - 1, 2, . . . , t hen A a, ¢ 0 ) if and only if #e is a -smooth on a(£). Thus there

exists a one-to-one correspondence be tween all regular zero-one a - smooth measures on a(£) and all ~-ultrafilters wi th the c .m.p .

2 . - P r i m e £ - f i l t e r s a n d z e r o - o n e m e a s u r e s .

IJE2Cil~IA 2.1. -- I~et # be a zero-one measure on :R(£), and let F = ( a ~ £: #(a) ---- 1) Then /7 is a p r ime S-filter.

Conversely, g iven a p r ime S-filter F , can we associate a zero-one measure /,~

wi th it? I J em m a 1.2 tells us t h a t we will not be able to define a n y s u c h / , ~ on all of A(F) . One way of proceeding is g iven in SI]~O~SKI [19]. However , we will p rov ide here the following shor t direct proof : for a n y a e ~, define

1 if a e F,

[ o if a~F.

150 I~O:NALD COHEN: Lattice measures and topologies

Then fi~ is a zero-one lattice function which is monotone and finitely additive. Le t ~ ( £ ) = {aAb': a, b e t } ; 3¢(£) is the semi-ring generated by £. Wi th no loss

of generality, we can always assume tha t b <a . Define fix on 3¢(£):

t@(aAb ) -- fi~,(a) -- fie(b) =

L ~ A 2.2. - a) fly is well defined;

b) fl~ extends fix ;

e) fir is finitely ~dditive on 5¢(£).

1 if a e F , b ~ ,

0 if a, b ~ ,

0 if a, b ~ .

P R O O F . - We shall omit the proofs of a) and b) and only prove e). Suppose tha t

aAb'=Va#Ab~, where (asAb~} is a disjoint collection in 3¢(£). 1

Case 1: #F(aAb')=O. For any j , we have a#Ab~<a/'\b'. Suppose tha t fl~(%Ab~)= 1. This implies t ha t a#eE ~nd b ~ F . As o = aAb'Aa~>~%Ab~Aa ', we have tha t a~.<aVb#. As a~eF, aVb~eE. As b # ~ F and F is prime, we have t h a t a e F . Similarly, one proves tha t b ~ H. But then we have the contradiction tha t flv(aAb') = 1, so our assumption mus t be wrong,and flx(aj'xb~)--= 0 for every j ;

~b

hence ~,flx(a~A b~) = O. 1

Case 2: Suppose tha t fiF(aAb')-~ 1, but tha t ~#v(a jAbj) -~ 0. Then for some 1

integer k~ a~,...,ato, b~, . . . ,b~eE and a~+~,...,a~, bT~+~,...,b,~F. Then we have

tha t a-~ (aAb')Vb= a;Ab#Vb, and therefore 1

n n /¢ k n

aAA b;----V (a~Ab;A A b~)VbAA b~<Va~Vb 5 = 1 ~ = / ~ + 1 ~ = i ~ = i I C + I

which implies t ha t either b e F or some a~ e _F (where j > k), in either case a contra- diction.

A f There is no th i rd case, as it can easily be shown tha t ~/@(ajAb~)<1.

1

IJE~51A 2.3. - Le t :K(g) be the class of finite disjoint joins of elements of JC(g): ~(g)

! A ! is the ring generated by g. Define/~v on N(£) by /zF(Va~Abi ) - ~/~(a~Ab~). Then 1 1

a) #~ is well defined;

b) #v is addit ive on :K(~);

e) #~ is the unique extension of ~x.

t~O~ALD COHEI~: Lattice measures and topologies 151

PROOF. -- This theorem is well-known: a proof m ay be found in TAYLOR, [22]. We h~ve proved

THEOlCEN[ 2.1. - There exists a one-to-one correspondence between all zero-one

measure on ~(£) and all pr ime G filters.

We would now like to find an analogue of theorem 1.3: namely, necessary and sufficient conditions on a pr ime filter F such t h a t / , ~ is a-smooth on ~(£). In section one we could restr ic t ourselves to the behavior of F and # on £ (due to the maxim~ll i ty of F and the regular i ty of #). ~ o w the i r behavior is more in t imate ly re la ted to Je(£), which makes invest igat ion much more difficult. The following example, which shows t ha t countable addi t iv i ty is not equivalent to a-smoothness (on Je(£)), i l lustrates the awkwardness of working on Je(£).

EXAMPLE 2.1. -- Le t P---- the semi-ring of half-open intervals on the real line of the form (a, b]. Le t Q = the rat ionals in (0, 1]. Le t P Q = the semi-ring of sets of the form (a, b] (~ Q. Define # on Po b y

b - - a if 0 < a < b < l ,

#{(a, b] (3 Q} = 1 o therwise .

Then # is a a-smooth measure on PQ which is not countably addit ive.

LE~I~A 2.4. - Le t flF and Je(£) be define as in lemma 2.2 ; if flF is countab ly addi- t ive, t hen #~. is a-smooth on ~(£).

PROOF. - This again is a well-known result , a proof of which m a y be found in TAYLOR [22].

Suppose t ha t Je(£) contains <~ sufficiently m a n y elements t h a t are regular with respect to a zero-one measure/z , . ~. To precisely define this s ta tement , let K = {x e ~ ( E ) : x is £-regular with respect to tt~, i .e. , / t~ (x)= sup/x~(a)}. W h a t we then are asking is w h a t happens if Je(£)c K~ ~<~

LE~)IA 2.5. - I f Je(£)C K, t h en #~ is £-regular.

PROOF. - P ick a e £ with # ( a ) = 0 . Pick any b e £ such t h a t # ( b ) - ~ l . Then #(aVb) = 1, (aVb)Aa'~Je(£), and tt~[(aVb)Aa']= 1. Since JC(£) o K , there exists c e £ , #~ (c )= 1, and c<(aVb)Aa'<a' , and so #~ is ~-regular.

The above hypothesis is actual ly a generalization of a s tandard definition:

DEFI~ITIO~ 2.1. -- A zero-one measure # on ~(£) is said to be £-tight if for any !

ai <a~, ai , a,@ ~, ~(a~) -- ~(ai) ~--- sup{#(b): b <a2Aa i, b e ~}.

TEEORE~ 2.2. -- A zero-one measure on a ring is C-tight if and only if it is C-regular.

152 ~:~ONALD COHEN: Lattice measures and topologies

P~oor . - One half we have Mready done, and the other half is tr ivial .

Given a lat t ice ~, define ~ ' ~ {x': x e ~}. I t is easy to show t h a t £' is a lattice, e e ~', and t h a t o ~ g~ if and only if e ~ ~. I t is also easy to show t h a t gO(g) = ~(g ' ) . Because of this relationship, if we s ta r t with a zero-one measure # on :R(g), we can associate with i t two different pr ime filters: / 7 : {ae £: # ( a ) : 1}, ~nd ~ ' ~ - { a ' e g': # (a ' ) -~ 1}. Another way of writ ing 17, is / 7 ' = { a ' e £ ' : a ~ / 7 } . Hence by working with one measure on ~(£) we can avoid working with two different filters on two different latt ices, as Frol ik does in [11]; however, we can still get his results with less labor. As an example, it is now easy to see tha t

C0~0LLA~¥ 2.1. - /7' is a (prime) filter if and only i f / 7 is a pr ime filter.

PROOF. - - ~Up, = ~ .

At this point in the paper , to make mat te r s easier, and wi thout loss of essential generali ty, we shall make the assumption tha t unless noted otherwise, all latt ices contain the maximal e lement e of the Boolean algebra. Hence f rom now on we use ~(~) and a(~) in terehangeably; t h e y are equal.

c o

DI~FINITIOI~ 2.2. -- An E-filter /7 is called g-Cauchy if given e-~Va~ , a~e~, ( n = 1 ,2 , . . . ) , t hen some a ~ / 7 . 1

Equivalent ly , we say tha t ~ zero-one measure # on ~(g) is g-Cauchy if given cO

e-~ Va~, t hen #(a~)----1 for some n. 1

LEPTA 2.6. - :lSet F be a pr ime g-filter.

1) /7 has c.m.p, if and only if /7' is £ '-Cauchy;

2) /7 is ~-Cauchy if ~nd only i f / v , has c .m.p.

P~ooF. - 1) /7 has e.m.p, if and only if o=Aa,~ implies some a ~ F , which is 1

equivalent to o --~/~a~ implies t h a t /zF(a~) ~ 0 for some a~, which in t u rn is equi- T !

valen t to e -~V a~ implies t h a t /~,(a',) = [/z~,(a~)] = 1 for some a~, which is t rue if and only i f / 7 ' is g'-Cauchy.

2) /7' has c.m.p, by (1) is equivalent to F " : F is GCauchy. Suppose tha t #~ is a a-smooth zero-one measure on a(g). Then b y definition F

has c .m.p. ; since / ~ / ~ F , we see t h a t F has c .m.p . , and b y the last theorem F is £-C~uchy. Hence a necessary and sufficient condit ion on ~ f o r / ~ to be a-smooth would have to be s tronger t han both c.m.p, and g-Cauchy. Such a condit ion is given below.

D]~FIlVITIO~ 2.3. -- An E-filter F is said to be s t rongly GCa.uchy if whenever given ~ o ¢ o !

e = V a~VV b~ (all of the a~ and b~ belong to ~), t hen ei ther some a~ e F, or some b~ ~/7. 1 1

RONALD COH~,N: Lattice measures and topologies 153

T~E0~EM 2.3. -- Le t F be an g-filter. /v is s t rongly g-Cauchy if and only if #~ is a-smooth on a(g).

PROOF. -- I f / t~ is a-smooth, t hen it is clear t ha t F is s t rongly g-Cauchy. Suppose/~ is s t rongly g-Cauchy. I t is immedia te ly obvious t h a t F is pr ime. In view of lemma 2.4~ i t is sufficient to show t h a t / t F is eountab ly addi t ive on ~(g) .

We proceed in the same manne r as in the proof of lemma 2.2c. Suppose aAb'-= ¢ o !

~-V ajAbj eJC(g), where {a~Ab~} is a disjoint collection. 1

Case 1: Suppose/tF(aAb')--~ 0. Then again, b y monotonic i ty , we are done.

Case 2: Suppose that / tF(aAb' ) - -= 1, bu t t h a t ~/t~(a~Ab~)= 0. We can assume 1

t h a t al~ ...~ a~ bl, . . . , b~ ~ F, and a~+i, . . . , bk+l~... 6 F (k m a y be infinite). Then we /c o o ]z c o !

get t h a t aAAbj<V ajVb, which in t u rn implies t h a t e = V a~Vb\/a!VV bj. But no I ~=/c+l I k+l

aje_~; also b ~ F . And a,b~,...,b~eF. But this is impossible if /v is s t rongly

g-Cauchy, and we are done.

As we have ah 'eady observed, if a f i l ter /~ is s t rongly g-Cauchy, t h en i t is g-Cauchy and has the c.m.p. Le t us define a filter to be b-complete if it is dosed under coun- table meets . We have the following result :

CO~0LLARY 2.2.-- Le t g be a b-lattice. I f F is s t rongly £-Cauchy, t hen F i s b-complete.

PROOF. - Let (a~} c F , and let a = Aa~. /t~ is ~-smooth~ so/tF(a) -~ 1. As g is a (~-lattice, a e £; hence a ~ F . 1

We now invest igate the propert ies of a v e ry impor t an t class of lattices~ those which are complement generated.

DEFINITION 2.4. -- We say t h a t g is complement genera ted if for every a e g, o o

a = Aa~,, where as e g for eve ry n. 1

THEOREM 2.4. -- Le t g be complement generated, and l e t / t be a zero-one measure on a(g). I f / t is g-Cauchy, t hen # is g-regular.

PROOF. -- Le t F be the associated g-filter; we will show t h a t / g is an ultrafi l ter . c o l c o

Pick a e g , and suppose a~F. As a=Aa.~, we have e----aVVa~. As / t ( a ) = 0 , 1 1

some a~ has / t-measure one. Therefore a~<a', and b y corollary 1.1, we are done.

COROLLARY 2.3. -- Le t g be complement generated. If/t is a zero-one a-smooth measure on a(g), t hen # is g-regular.

15~: ]~ONEL]) CO~EN: Lattice measures and topologies

W e can even do be t t e r :

TttEORE~:[ 2.5. -- :Let g be complemen t genera ted. Then # is a -smooth if and only

if # is g-Cauchy.

P~oo~. - One half we have a l ready done. Suppose t h a t # is g-Cauehy. We will c o c o c o

f

show t h a t # is s t rongly g-Cauchy. Given e = V a~\/V b~. Each b~ = V b~k, so t h a t oo oo i 1 i

e-----Va~V v b~. Now either some a~ has ~-measure one, or for some bi~ ~(b~)-~ 1 i,~=l

=/~(b~) = ] .

At this point we can complemen t a resul t of SP~ED [21]: in l e m m a 2.5 be s ta tes t h a t if g is a d-latt ice which is complemen t genera ted and no rma l (see definit ion 3.5c),

t h e n if ~ is a (%filter, i t is an g-ul tra filter.

COROLLARY 2.4. -- Le t g be complemen t genera ted , and F a p r ime g-filter. I f

F is £-Cauchy, it is a &comple te g-ultrafilter.

3. g-filters and zero-one measures; separation axioms.

DEFINITION 3.1. -- :Let tt and v be two zero-one measure on a(g). I f for every x e g

it is t r ue t h a t tt(x)<v(x), t hen we wri te # <<v(g).

Observe t h a t if # is g-regular, t h e n # < v ( g ) implies t h a t # = v.

DErI~ITION 3.2. - I f A c g is an g-filter base, we denote b y [A] the g-filter gene- r a t ed b y A.

THEOREN 3.1. -- I f A = {a~} c g has the Lm.p. (i.e., eve ry finite mee t of ele-

men t s in A V: 0; equivalent ly , A is an g-filter base), t hen there exists a zero-one g-regular measure # on a(g) such t h a t t t ( [ A ] ) = 1.

PRooF. - [A] c F an g-ul traf i l ter ; #F([A]) = 1.

COI~OLLAI~Y 3.1. -- L e t tt be a zero-one measu re on a(fi). :Let b e g be such t h a t

/~(b)= 0, bu t for a n y a E g such t h a t #(a)= 1, aAb#O. Then there exists a zero- one measm'e v on a(g) such t h a t /~<v(g), and v(b)= 1.

PROOF. - :Let f = {a e g: / t (a) = 1}, and let V = [ F U {a}].

As we shall see, i t is possible to impose condit ions on a la t t ice t h a t can be viewed

as ve ry general abs t rac t ions of separa t ion notions of topology. However , because of its length, we shall consider elsewhere a more detai led analysis of la t t ice separa- t ion, and shall only in t roduce the m i n i m u m n u m b e r of condit ions needed for this paper .

I~ONALD CO~E~: Zattice measures and topologies 155

D E ~ I O ~ 3.3. - ~ is called basic if given an y x=/=e, xeg , the re exists an a tom b e $ such t h a t b<x'.

DE~Im~O~ 3.4. - Le t ~ be ~ sub-latt ice of an y algebra A. Two zero-one measures # and v on A are 35-compatible (wri t ten #~ , , ( JS) ) if whenever a, b e ~ s~ch t h a t #(a) ~ v(b) : 1, t hen nab =/-= O.

LE)[~¢~ 3 . 1 . - Le t /z~ and ~u~ be zero-one measures on a(JS). I f #~/z~(JS) , t hen there exists a zero-one ~- regula r measure v on a (~) such t h a t / ~ <v(~) , a n d / ~ <v(YS).

PRoof . - Le t M - ~ { x ~ : # ~ ( x ) ~ l or ~ ( x ) : ] } . M has the f .m.p . , so let be the J~-regular measure described in theorem 3.1 t h a t is one on [M].

DE~I~m~o~ 3.5. - a) g is T~ if it is a tom relating, 55 is atomic, a.nd if whenever given two dist inct atoms a, b ~ 55 there exist x~ y e £ such t h a t a < x ~, b ~ x', and b < y ' , a ~ y ' .

b) ~ is said to be Haasdorf f l

a~, a ~ such t ha t b~<a~, b~<a~, if whenever given atoms b~ v~ b, ~ 55, there exist

! !

and a~/~a~-----O. I f in addi t ion g is a tom relat- ing and 55 is atomic, we say t h a t £ is T~.

e) We say t h a t C is normal if for all a, b ~ ~, aAb= O, the re exist e, d ~ such t ha t a<e', b<d', and e'Ad'= 0. A normal a tom disjunctive lat t ice which is T1 is called T4.

In view of definition 3.4, it is t r ivial to show th a t a lat t ice ~ is t tausdorff if and only if for all a toms b I ve b2, ~ub~/~b~(U ). TOt So tr ivial is the following theorem.

THEORE~ 3.2. - ~ is Hausdorff if and only if for each zero-one measure # on a(~), there exists at most one a tom bE:5 such t h a t for all xe~, # ( x ) : 1 implies x>b.

PnooF. - Suppose ~ is t Iausdorff , bu t suppose t h a t there exists a zero-one mea- sure # such t ha t for all xE~ , if # ( x ) : l , t hen x>biVb~, where b~ a n d b~ are

f ! unequal atoms. Then there exist c~, c~e ~ such t h a t b~<c~, b2<c2, and c~/c2-~ e; bu t #(Cl)= 0 as e~Ab~: O, and # ( e 2 ) : 0 as b~Ac~: 0; contradict ion, and we are done.

To prove the converse, we will make use of the r emark preceeding this theorem. Suppose t h a t we do have our unique a tom, bu t also there exist a toms b~ ¢ b~ with /~b~ ~/~b~(U) • B y lemma 3.1 there exists a zero-one U-regular measure v such t h a t /~<v(~ ' ) i ~%,<~(U). P ick any d e ~ such t h a t v(d) ~ 1. Then ~(d') : 0 implies t h a t #b,(d ~) ~ #~.(d ' )~ 0, which implies t h a t #~(d)- : - -#~(d)= 1, which in t u rn implies t h a t b~<d and b~<d, con t ra ry to our assumption.

We get ~s a corol lary a r e s e t of : F ~ o ~ ; [1:i]:

COrOLlArY 3.2. - Le t ~ be a tom relating, :~ atomic. Then ~ is T~ if and only if for all p r ime filters F ,

x ~ F } = I b an a tom, o~, A{x:

t O.

156 ~ONALD COHEN: Lattice measures and topologies

THE01CE~ 3.3. - £ is no rma l if and only if for all t r ip le ts #, v, y of zero-one mea-

sures on a(£) (where v and y are £ - reg~ar ) , i i # < v ( £ ) and # <},(g), t hen ~ ~.

PROOF. - Suppose t h a t £ is normal , suppose # <v(g) and ~t <y(g) . Then we claim t h a t y ~---v(£). For if not~ then the re exist two disjoint e lements of £, say a and b,

wi th v(a) = 7(b) = 1. Then there exis t c', d' separa t ing a and b, with cVd----- e." Hence

ei ther c or d mus t have # -measure one. I f v(c) ~-- 1, t hen v(aAc) = v(o) = 1, which

is impossible. Therefore ~'(c) = 0 r so #(c) = 0, and hence/~(d) ~ 1. B u t t hen y(d) -~ t which implies t h a t y(dAb) = ?(o) = ] , which is absurd. Therefcre # and v are £-com- pat ible . B y l e m m a 3.1 there exists a zero-one £-regular measure i such t h a t v <),(g) and y < / ( £ ) . Bu t v is £-regular implies t h a t v ----- i , and s imilar ly ? = t . Therefore

y ~ v .

Conversely~ suppose t h a t t he hypothes is on the measures is t rue , bu t g is not

normal . Then there exist a, b e g such t h a t if G = {x' e g': a <x '} , H = {y' e £ ' : b <y '} , t h e n G U H has the f .m.p . B y t heo rem 3.1 there exists an £ ' -regular measure v

such t h a t v ( [ G W H ] ) - ~ 1 . Le t y e £ ; i f v ( y ) = 1 and y A a = 0, t h e n y ' > a , which im- plies t h a t v ( y ' ) = i , which is impossible. Therefore if v ( y ) = 1, y A a # O . Bycoro l -

la ry 3.1 there exists an £-regular measure t~ such t h a t v</~(£) . Similarly, we show

t h a t the re exists an £-regular measure i~ such t h a t 2~>~v(£). B u t ~ ( a ) - = ).~(b)= 1,

so 1~=/:~, contradict ion. Therefore £ mus t be normal .

We get as a corol lary a resul t of F~oLIK [11]:

COROL]~A~Y 3.3. -- £ is no rma l if and only if given a n y three p r ime £-filters /~,

G, H (G and H are ultrafi l ters) wi th F c G and F c H, t h e n G = H.

DrINITIONE 3.6. -- a) I n ( £ ) = {#: # is an g-regular zero-one measure on a(£)}.

b) I ( £ ) = {~: # is a zero-one measure on a(£)}.

We recall t h a t the s y m m e t r i c difference of a and b, a A b = (aAb')V(bAa') .

DEFINITION 3.7. -- Le t x~yea(£) . We say t h a t x = y a lmos t everywhere (a.e.)

if for every #elR(£)~ # ( x A y ) = 0 . I f x # y (a.e.), we say t h a t In(£) separa tes x

and y.

DEFINITIO~ 3.8. -- £ is called pairwise d is junct ive if for all a, b e £, a ~ b, there

exists de£(d=/=O) such t h a t d < a and d a b = O.

THEOI~E~ 3.4. -- £ is pairwise dis junct ive if and only if for all a, b e £, a = b (a.e.) implies a = b. I n such a case we say t h a t I~(£) is separa t ing .

PzooF. - Suppose £ is pairwise disjunctive. Suppose we have a, b e £, and wi thout loss of general i ty , a ssume b $ a. Then the re exis ts c e £ , a A c = O and c<b. Then there exists # e I~(£) such t h a t ~t(c) = #(b) = 1 and #(a) = 0. Then #(a A b) = 1, and

so a:#b (a.e.).

]:~,ONALD COttElg: Lattice measures and topologies 157

Conversely, suppose Ig(~) is separating, and pick a, b e ~, a C b. Then there ex- ists /~.elg(g) such tha t , wi th no loss o£ generali ty, # ( b A a ' ) ~ 1. Then clearly b ~ a and b y regular i ty there exists c e ~ such tha t # ( c ) = 1, c < b A a ' . Bu t then c < b , and we are done.

E X A ~ L E 3.1. -- As ~n example of ~ lat t ice thg t is not p~irwise disjunctive, pick any infinite set X and give it the p a r t i c ~ a r point topology. (There is one point p e X such t h a t a set 0 is open gnd only if p c 0 , or 0~--- 0.) I f g : t he open sets, t h en ~{p} ca(g) , ~nd ~{p} ~--0 (a.e.), as for ~ny # e I ~ ( g ) , #(~{p})~--0. More on this ex- ample later .

LEMMA 3.2. -- I f g is pairwise disjunctive, then for eve ry x e ~(g) (x ¢ 0) there exists ae£(a¢O) such t h a t a<x.

l

P]~OOF. - w = Va~Ab~ >alAb 1. 1

cabs= O. Then c < x .

a ~ b ~ , so there exists e ~ g such t h a t e<a~,

Again this is not t rue for an a rb i t r a ry lat t ice. Wi th the same topology as in ex- ample 3.1, ~(p} e ff~(g), bu t there does not exist any a e g wi th a<~{p}.

COrOLLArY 3.4. - I f g is pairwise disjunctive, t hen for all x, y e fit(g), ~---- y (a.e.) if and only if x ~ y.

4. - W a l l m a n - F r i n k type topologies .

I n this section we shall topologize the zero-one measures on a lat t ice. These topo- logies have been looked at f rom an ultrafi l ter viewpoint by SAMUEL [19], and f rom a pr ime ideal point of view by Gn£TZE~ [14] and others.

We use as a base for the closed sets of a topology on In(E ) sets of the fo rm W(a) -----{ueI~(~):l~(a)--: 1}, where a e g. We denote this topology by ff)~. I t is easy to show thu t ~D~ is TI:

THEOlCE)I 4.1. -- (I:a(~), ~)u) is compact .

PROOF. -- We will show tha t any collection of basic closed sets with the finite intersect ion p rope r ty has non-empty intersection. Le t K----- {W(a~)} have the f . i .p. , and let / 7 = {a~e~: W(a~)eK}. I f a~, a~EF, t hen W(a~Aa~)~ W(a~) n W(a~) V=O. which implies t h a t a~Aa~#0 , so t h a t F has the f .m.p.

Le t [F]cG an £-ultrafilter. # G ( [ F ] ) = 1 , whence /~ac(~W(a). Consider now the class %0 of basic closed sets in ~ : %0 = {W(a): a e g}. %0 is a

lat t ice under unions and intersections, for W(a) ~ W(b) -~ W(aAb), and W(a) L) W(b) = W(aVb). We now exhibi t a na tura l isomorphism between a(g) and a(%0). Define,

1 1 - A~naI i di Malemat ica

158 ~ONALD COHES~: Zattice measures and topologies

for xea(£), T(x)~- {#eI~(£) : # ( x ) = 1}. I f Uea(2D), t hen U = 0 [W(a~) (~ ~W(b,)], n " i = I

where all of the a~, b, e t . I f x=VaiAb~ea(£) , t h en U-----T(x).

LEM~A 4.1. -- I f £ is pairwise disjunctive, t h en for any a, b e ~, W(a) = W(b) if and only if a = b.

PROOF. - Suppose a, b e ~, and wi thout loss of generali ty, assume t h a t b ~ a. Then there exists d e s such t ha t d<b and a A d = O. Hence we can find a zero- one ~-reg~lar measure # such tha t # ( d ) = 1; therefore # ( b ) = 1. Bu t # ( a ) = 0, so

w(a) =/= W(b).

In light of lemma 3.2, we can easily ex tend this to all of a(£).

THEORE~ 4.2. - I f ~ is pairwise disjunctive, t hen for an y x, y e a(~), T(x) = T(y) if and only if x = y.

Assume for the diseusion tha t follows tha t ~ is pairwise disjunctive. We can now exhibi t a na tura l correspondence between I(£) and I(%0). For any

/~eI(£) a, nd for any xea (~ ) , let f i (T(x))=#(x) , fi is well defined, for T(x )= T(y) if only if x = y. I t is t r ivial to show tha t fi is a measure. The map # -+ fi is one- to-one and onto; hence we have.

T~EORE~:[ 4.3. - - I f ~ is pairwise disjunctive, there exists a one-to-one correspon-

dence between all zero-one measures on a(~) and ~11 zero-one measures on a(ql~).

Suppose t ha t /~ e Ig(~); t hen even if ~ is not pairwise disjunctive the map/~--> fi is still well-defined for if T(x) = T(y), t hen b y definition #(x) ----- #(y). ~ o t only is the map well-defined, bu t it is easy to show tha t fi is %0-regular. Hence we have a corollary.

COROLLARY 4.1. - There exists a one-to-one correspondence between Ig(~) and ~ ( ~ ) .

For the following lemma we shall use several (i topological )) results about lat- tices, the proofs of which are quite direct.

ZEMMA 4.2. - I f (I~(~), ~)~) is T~, t hen ~ is 1'4.

P R O O F . - - Since D~ is T2, any two points/~, v can be separated by disjoint basic open sets: hence ~1) is 1"2. ID~ is compact , so any covering b y open sets (in par t icu lar a ny covering by basic open sets) has a finite subcovering, so t h a t ~ is compact . Hence ~l) is normal.

T~EOREM 4.4. - Suppose ~ is parwise disjunctive. Then £ is normal if and only if (I~(£), ff)~) is Hausdorff .

RON_~,D CO]~EN: Lattice measures and topologies 159

PROOF. - - Suppose £ is normal and t h a t /~ ~ v; t h en there exists a, b e ~ such t ha t # ( a ) = 1~ # ( b ) = 0, and v(a)----0, v (b )~ 1. v is ~-regular so there exists e<a ~ such t h a t v ( c ) : l ; wi thout loss of generali ty, assume t h a t c--:b. Then aAb-~O, and by normal i ty there exist [ , ge £ such t h a t a<['~ b<~g', and ] 'Ag'= O. Then # ~ °W(]), ~, ~ °W(g), and °W(]) (~ °W(g) = 0.

Conversely, suppose t ha t ~ is T~, and suppose there exists a zero-one measure # on a(C) and two £-regulax measures v~ and v~ such tha t # <~,~(~) and # <v~(~). Con-

sider now fi and gl. I f for any W(a)s ~ we have p(W(a))= 1, t h en /~(a)= 1, which implies t ha t r~(a) = 1, which in tu rn implies tha t g~(W(a)) = 1, so tha t fi < 4~(21)). Similarly fi <~(~t)). By temm~ 4.2~ %0 is normal , so t h a t by theorem 3.3, ~ = g~; bu t t hen v~= v~, and we have p roved t h a t ~ is normal.

THE0~EM ~.5. -- Suppose tha t ~ is basic and a tom disjunctive. Then in the O - topology we have :

a) {#b:b is an a tom of :5} is dense in IR(£).

b) The mapping taking each a tom b into /~b is one-to-one.

PRooF. - a) Le t # s I~(C), and pick a basic open neighborhood ~W(a)D ~u}, SO t h a t #(a ') ---- 1. ~ is basic, so there exists an a tom b <~a'. Then #b(a') ---- 1, so #b s ~W(a).

b) The proof is s t ra ightforward and will be omit ted.

We can now apply this f ramework to a complete ly regular T1 topological space. As proved by FRI~K [10], such a space always has a base ~ for the closed sets which is a I'4 latt ice. According to theorem 4.4, ~)~ is Hausdorff . The mapping defined in theorem 4.5b can be shown to be g homeomorphism.

COROLLARY 4.2. -- Le t £ be a T1 dis junct ive base for the closed sets of ~ topologi- cal space X. Then IR(C ) with the ff)~ topology is a T1 compactif icat ion of X. ff)~ is T2 if and only if £ is T~.

EXAMPLE 4.1. - Le t X be a Tychonoff space.

a) Consider the latt ice 3 of zero-sets of continuous funct ions; in this case

I~(3) ~ fiX, the Stone-Ceeh compactif icat ion of X. 3 is /'4, so f i x is T~.

b) Consider the lat t ice C of all closed sets. I~(C)----~oX, the Wal lman com- paetification of X. By theorem 4.4, o)X is t tausdorff if and only if X is normal.

EXAMPLE 4.2. -- Le t X be an ul t ra-regular T~ space (it is a regular space where disjoint points and closed sets can be separa ted b y disjoint elopen sets). Le t 8 ~ the clopen sets. 8 is ve ry easily seen to be T4, and in this instance 12(8 ) ~ fleX, t he Banaschewski compactif ieat ion of X.

160 ~ : ~ O N A L D CO]~E:N: Lattice measures and topologies

$. - L a t t i c e r e l a t i o n s h i p s .

For most of this section we will invest igate the relat ions between I~(£~) and I~(£~), where £~ c £~.

DEH~ITION 5.1. - Le t £1 c £~. I f v e I(£~), we denote it res t r ic t ion to a(£1) by ~1£~.

L~A 5.1. - Given £~ c £=. To every ff e In(£~) there corresponds v e I~(£~) such that ,u = vI£,.

P~oo~. - Le t ~ = {me£~: there exists ae£1 such t h a t • a ) = 1 and a < x } . J~is an £~-filter, so there exists v e I~(£~) such tha t # <v(£1). I t remains to show tha t

# = ,, on £1, and hence on a(£1).

Le t a e£1 , v ( a ) = l . I f # ( a ) = 0 , t h e n ~s # is £1-regular, there exists c~£1, c < a ' , and # ( e ) ~ 1. Bu t t hen v ( e )= 1, and then v ( a A e ) = v (o )= 1, which is im-

possible. Therefore i f ( a ) = 1, and # = v]~.

DEFINITION 5.2. -- a) We say t h a t £1 semi-separates £~ if for all b e £1 a ~ £~, a A b = O, the re exists ee£1 such t h a t a <c and c a b = O.

b) We say t ha t £1 separates £~ if for all a, b e £3 such t h a t aAb = O, the re ex-

ists f, g e £1 such t h a t a<], b<g, and l a g = O.

c) We say t h a t £1 coseparates £~ if for all a, b e £3 such t h a t aAb = O, there exist e, d~£1 such that a < c ' , b < d ~, and e'Ad'-~ O.

THEORE~ 5.1. -- Given £1 c £~. Then the following conditions all imply t h a t £1 semi-separates £~: (a) £1 separates £~; b) £1 coseparates £~; c) £1 is complemented.

The reason for the in t roduct ion of semi-separation becomes clear in the follow-

ing lemma.

LEMI~± 5.2. - - Given £1 c £3, suppose t h a t £1 semi-separates £2. Then for every

P~oo~. - I t is t r ivial to ver i fy tha t vI£1 is a measure. To show tha t it is £1-regular, suppose t h a t there exists a~ £~, v ( a ) = 0. Then v ( a ' ) = 1 implies t h a t there ex- ists be£2, b<a' , with v ( b ) = l . But t hen there exists c~£1 such tha t b < e < a ' , with v(e) = vl~(c ) = I.

Combining lemmas 5.1 and 5.2, we have shown that there exists a map, with

we shall denote by ~, which takes ~ll of I~(£~) onto I~(£,): for v ~ I~(£~), ~(v)= vl£ ~.

Unfor tuna te ly , ~b need not be one-to-one. We r emedy this by defining an equiva- lence relat ion on I~(£~):vl~r~ if and only if v~[£ =v2I£ . As it tu rns out we have seen this equivalence relat ion before: v~--~v2 if and only if vl~v~(£1). To prove this, suppose t h a t vlI£ =v~I~,, where vl,v~elR(£~ ). I f for a, b e £1 we have t h a t

I:~OINALD COHEn: Lattice measures and topologies 161

v~(a) = v~(b) = 1, then also v~(b) = v2(a) = 1, and so a A b # 0 . Conversely, if v~_~ v~(C,) t hen by lemma 3.1 there exists a zero-one C~-regular measure 2 such tha t v~]C ~ ~< %(~), and v~l~ ~< 2(~). As both restrictions are ~l-regular, we have t h a t the two restrictions are both equal to Z, and hence are equal.

Denote by I ~ ( ~ ) / ~ the set of all equivalence classes, and denote a t y p i e a l class element by ~. Define the map ~ to be the inclusion map taking v e I~(~) into ~, and define ~5(~) = v]~.

THEOREM 5 . 2 . - Given ~ c Cz and suppose t ha t C~ semi-separates C~. Then ¢ : (I~(~), il)~)-+ (I~(Cx), ~ ) is a continuous surjection.

P~oo~. - We need only show show continuity. Zet Wl(a), a e ~, be a basic closed set in I~(~) . Then q~-i(W~(a)) = zc-~(~5-~(W,(a))) = 7¢-~{~: vice(a) -~ 1} ----- {veln(£~):

= = W (a) c

EXAMPLE 5.1. - Let X be an ultraregular T~ space.

a) Ze t C1= 8 = clopen sets, and let £3= ~ = the zero sets ( = 6(£~), the smal lest latt ice containing ~x which is closed under countable meets). Then 8 c 5 and 8 is complemented; by theorem 5.1e, $ semi separates ~. Hence there is a continuous mapping f rom I~(8) (= fiX) onto I~(5) ( = floX).

b) I~et g l = 8 = elopen sets; let £ ~ ~ ( 8 ) ~ the closed sets. Then there is a continuous surjeetion from I~ (~ (8 ) )= c~X onto fioX.

THEOREM 5.3. -- Given CI c £~. I f ~ separates or coseparates g~, then q~ is one- to-one. (Itenee if £1 is normal, I~(~) is T2, and qi is a homeomorphism.)

PROOF. -- Suppose ~1 separates £3, and pick v~ #v~ from I~(~). Since they are not equal, there exist b, eeC~ such t h a t ~ ( b ) = l and v2(b)=0; ~ ( e ) = 0 and ~(e) = 1 (and bAe = 0). Then there exist ], g e ~1 such tha t b ~<], e<g , and gA1----- 0. Whence v~(]) = 1, ~(g) = 0; ~,~(g) = 1, and v2(]) = 0, which shows t h a t vl]~ #v~]~.

Now suppose t ha t £~ coseparates ~ . Given b and c as above, this t ime there exist h, k e ~ such tha t b<h' , e<~k', h'Ak'==O, so t h a t h V k - ~ e . Now b A h = O , so t h a t r l ( h ) = 0, whence ~ ( k ) = 1; also e A k = 0, so tha t v2(k)= 0, and therefore

v~(h) = 1. Therefore v~tc~:/:~[z~.

THEOREM 5.4. -- Suppose t ha t £~ c C~ and tha t ~1 semi-separates ~ . Then the map ~b is one-to-one if and only if ~ separates ~ .

PROOF. -- We have proved one half in the previous theorem. Suppose therefore t ha t ¢ is one-to-one, bu t t ha t there exists a disjoint pair a, b e £2 tha t cannot be separated by two disjoint, elements of ~1. I f we set

K = {xeCl:a<x} u {yeCl:b<y} ,

162 ~O:NALD COI-]::E:N: Lattice measures and topologies

then K h~s the ].m.p. ]Set 9 be the zero-one ~-regular measure described in theorem3.1 which is one on [K]. Le t z e £ ~ 9(z)---- 1. I f z / \a= O, then b y the semi-separat ion p rope r ty there exists c e ~ such tha t zAc-~O, and a4c~ so t h a t v ( c ) = l ; bu t t hen we arr ive at the impossible conclusion t h a t 9(0)-~ 9 ( a A c ) ~ 1. Therefore zAa =/:0. Similarly we prove t h a t zAb ~0 . With a slight var ia t ion of the proof of corollary 3.1 we can now prove tha t there exist two measures ~, 2 e In(£~) such t ha t ~ ( a ) = ~ ( b ) - 1, and ¢ ( ~ ) ~ ~b{~)~ 9, which is impossible if ~ is one-to-one. Therefore a and b can be separated, and therefore £~ separates £~.

CO~OLZAICY 5.1. - Given £~ c ~ . I f ~ coseparates ~ , t h e n £~ separates £~.

])ROOF. - Theorem 5.3 and 5A:.

We can apply our work to get a ve ry direct proof of the following (see e.g., [4])

CORO]~ARY 5.2. - Le t X be a normal, u l t raregular space. I f fioX= fiX, t hen X is n l t ranormal . (By theorem 5.3, if X is s t rongly zero-dimensional ~oX = fiX.)

P~ooF. - X is normal, so f iX= ogX; therefore wX = fioX, and hence the map O : I a ( ~ ( 8 ) ) - ~ I ~ ( 8 ) is one to one. 8 is complemented implies t h a t 8 semi-sepa- rates 3(8); hence by theorem 5A, 8 separates z(8): bu t this is the definition of ultra- normal .

Although we will not invest igate its propert ies now, if again £~ c £2, and this t ime £z is normal, t hen there exists a well-defined mapping 0 f rom I~(~.) onto In(~z) such tha t if 9 e I~(~) , t hen ~ <0(9)(~). To prove this, just note t h a t 9]c ~ is a zero-one m e , sin°e, and since ~ is normal, i t has a unique extension to a regular zero-one mea- sure on a(~) .

LEPTA 5.3. - Le t fi be T~, and consider the lat t ice ~(~). (~(~) is the smallest class containing ~ which is closed under a rb i t r a ry meets). Suppose also tha t ¢ : Ia(z(~)) --~I~(~) is well-defined. I f for 9 e/~(~(E)) we have ¢b(9) -~/~b, where b is an atom, then 9 = v b.

PROOF. - Le t G = {g ~ v(g): v(g)= 1}. For every g~, ~ G, g~- A{]~: ]~e ~}. Clear- ly for every c¢, fi we have 9 ( ] ~ ) = 1, and therefore #b(]~)~-1. Bu t t h e n

As each ]~z has/~b-measm'e one, each ]~>b, and so we see tha t G is fixed. ~ is T~ implies t ha t 3(£) is T~, and so b y theorem 3.2, we see t h a t v = v b.

DEFINITION 5.3. -- The set {b} of a toms of :5 is E-replete if each zero-one £-regular ~-smooth measure on a(£) is equal to some #b • In the special case where :5 = if(X) and ~ is a lat t ice of sets, we say tha t X is E-replete if each zero-one ~-smooth measure on a(~) is a /~b-

RO:~ALD COHEN: Lattice measures and topologies 163

I n the ease of a comple te ly regular T~ space the word comple teness is used b y

some au thors (see [13]). I n the W a l l m a n - F r i n k f r a m e w o r k (see [i0] and [25]), ALb and S~APmO [1] speak of ~-realcompaet where ~ is a no rma l base la t t ice in the sense of F~I~K [10]. Again b y su i t ab ly topologizing the la t t ice we can get sevei 'al resul ts

on repleteness . Below we give a direct proof of a resul t of DYKEs [8].

THEOREM 5.5. -- Le t X be a Tychonoff space. I f X is 5-replete, t hen X is C-replete.

PROOF. - Recal l t h a t 3 is complemen t genera ted . Suppose t h a t v is a a -smooth

zero-one C-regular measure on a(C). I t is i m m e d i a t e l y clear t h a t vI5 is 3-Cauchy a n d is thus , b y an appl ica t ion of t heo rem 2.4 and 2.5, a 5-regular a -smooth zero-one

measure on a(5). B y repleteness , we have t h a t v15= #~; b y l e m m a 5.3, v = vb, and

the re fo re X is C replete .

THEOREM 5.6. -- Given ~ c £~, where £~ is a t o m disjunctive. I f ¢ is one-to-one,

t hen ~- rep le teness implies £~-repleteness.

PROOF. - Le t ve/R(£~) be a-smooth . Then q b ( v ) = # e I ~ ( £ ~ ) is also a -smooth ,

whence # =/~b. I f £~ is dis junct ive, then % is an £~-regular zero-one measure , and

c lear ly ~b(vb)-= #b. As q~ is one-to-one, we m u s t have v = vb.

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