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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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[5] G. BACH~A~¢ - R. Com~¢, Regular lattice measures and repleteness, Comm. on Pure and Applied Math., 26 (1973), pp. 587-599.
[6] B. BANASCm~WSKI, O~ Wallman's method of compaeti]ieation, Math. Nach., 27 (1963), pp. 105-114.
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from Funktional'nyl Analiz i Ego Prilosheniya, 4, no. 3 (1970), pp. 51-60. [10] ~. F~I~;K~ C~m~a~ti~i~ati~ns and semi.n~mal spa~es~ Amer. J. Math.~86 (~964)~ pp. 6~2-6~7. [I1] Z. FRO~IK, Prime filters with CIP, Commen. Math. Univ. Carolinae, 13, no. 3 (1972),
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164 t%O~AI~D C0~E~: Latt ice measures and topologies
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