Compact, ifications Defined by Means of Generalized Ult~rafilters.

I~UGt-][ ~Ol~.DObI (Albang~ N.Y. , U.S.A.)

Dedicated to the sixtieth birthday of P.rof. Edgar R. Lorch

Summ:u'y. - Given a suitable ri~g of bounded real-valued function, s on a set X. the relation. ship between generalized ultvafillers a~,d the u~uol compactification deiermi~ed by the ring is studied. The classification of the ultrafilters as real or hyperreal is discussed.

The STONE-CECIl compactif icat ion of a completely regular space X may be constructed out z-ultrafi l ters. A z-ul t raf i l ter is a maximal dual ideal in the latt ice of zero sets of X, a zero-set of X being the set on which a bounded cont inuous real-valued funct ion vanishes. It is the ring C of all such func- tions which determines the STO~E-CEcg compactif icat ion, in the sense that this compact i f ieat ion is the only compact space Y on which the ring of con- t inuous real -valued funct ions is isomorphic to C.

In this paper we seek to generalize the const ruct ion of the preceding paragraph. We start with a set X and a r ing J~ of bounded functions on X; here and hencefor th << function~) means <<real-valued function>>. We may as well assume at the outset that there is a compact space Y such that J~ is isomorphic to the r ing of cont inuous funct ions on ]~. We can, and in fact

do, phrase this assumption in terms of conditions on ~ and proceed to con- s truct Y. However, it is not the exis tence of Y, which is well-known (see e.g. [5, p. 473] or [6, pp. 207-2(~9]), that is in quest ion here. Instead, we are con- cerned with the relat ionship between Y and the J~-ultrafilters. For the mo- ment, we may describe JB-ultrafilters as being ent i le ly analogous to the z-ul t raf i l ters of the preceding paragraph, except that we use the sets on which functions in JB, instead of continuous functions, vanish. We will call these sets J~-zero-sets.

An interest ing question is tile one of precisely why the J~-zero-sets, ra iher than their complements, which we call JB-cozero sets, are suitable for use in the construct ion of Y. In fact, the answer is that the cozero-sets will serve jus t

16 HUGH GORDON: Compactifications Defined by Means, etc.

as well as the zero-sets in the general case. The precise property enjoyed by the zero-sets in the continuous function case, which makes them work bet ter there than the cozero sets, will be found in Theorem 3.3. Incidentally, it will be seen that even in this case it is possible for the cozero-sets to have the property. Precise ly when X is an F-space. the cozero-sets and zero- sets will work equal ly well in defining the STo~]~-OEC~ compactif ication of X. On the other hand, in the general case, the zero-sets may lack the proper ty of Theorem 3.3 and hence again have no advantage over the cozero-sets.

As discussed by Lol~c~ [6], the classification of the points of the S~o~E- CECE eompactif ieation as real or hyperreal carries over into the general case. We show in section 4 how the two kinds of points may be dist inguished by means of the corresponding ~-ul t raf i l ters .

It should be pointed out the notion of a z-filter was introduced by HEWITT [3] and tlle term <<Z-filter>) by Ko~Ls [4].

The reader who is familiar with the book [2] by G:[LLMA:S and JEBISO~ may be puzzled at times as to why we proceed as we do. The reason we must sometimes take a different approach from that of the book is made clear by the following extremely simple example. Let X be the closed interval [0, 2] with the point 1 deleted. Let J~ be the set of restrictions to X of func- tions cont inuous on [0, 2]. Certainly the compact space :Y corresponding to the ring JB is [0, 2]. :Note that the half open intervals [0, 1) and (i, 2] are disjoint J~-zero-sets whose closures in [0, 2] are not disjoint. By contrast, we recall that in the C~ILLMAN-JERISO:N case of continuous functions, disjoint zero-sets have disjoint closures in the S~O~E-(~ECl=[ compactification.

1 . - Basic Notations and l,emmas.

1.1. - Let X be a set. We make the set of all bounded r e a l - w l u e d func. tions on X into a ring by introducing pointwise addition and multiplication. Let J~ be a subring of this ring. We suppose that JB contains the constant functions and separates points of X. We put :

llfll-- sup !f(x)} for each f ~ : ~ xi~X

and suppose J~ is complete in this norm. As is well-known, these hypotheses insure that J~ be a sublatt ice of the lattice of real-valued functions on X. Thus whenever f and g e : ~ , so are f V g - - s u I ) { f , g}, f A g - = - i n f If, g},

f + = f V 0 , f - = ( - f ) V o and I f l=f++f- .

1 . 2 . - By a J~-zero-set we mean a set of the form f-*[0) with f EJB. The complement in X of a JB-zero-set will be called a J~-cozero-set.

HUGH GORDON: Compactifications Defined by Means, etc. 17

I f f~, f~, ... e ~ , so a r e :

c o

and Z I f~i A (2-1e),

where e denotes the constant function 1. It follows that each finite union and each countable intersection of 5~-zero-sets is a ~-zero-set. It also follows that each finite intersection and each countable union of :~-cozero-sets is a ~-cozero-set. For each real number a and each fe~3, { x e X ] f ( x ) ~ o ~ t is the set on which ( f - - ~ e ) - vanishes and hence is a ~-zero-set. It follows that whenever ~ and ~ are real numbers and f e ~ , then f-l[~, ~] is a :~-zero-set, while f-l(~, ~) is a ~-cozero-set.

1 .3 . - Pr imar i ly in order to be able to deal with both the :~-zero-set case and the ~ cozero-set ease simultaneously, we introduce next a notion sufficiently general to include both. The important cases of what follows are the ones in which the $ we next define is either the collection of all ~-ze- re-sets or that of ~-cozero-sets. I t is also possible to choose for $ the collec. tion of all subsets of X; in this case, the ~- f i l te rs defined below are jus t the actual filters on X.

We select once and for all a collection S of subsets of X. We require that 8 be so chosen that

1) The intersection of each two sets in $ is in S.

2) For each f ~ , those neighborhoods N of zero, in the space of real numbers, such that f - l N e 8 constitute a fundamental system of neigh- borhoods of zero.

Noting that :~ contains the constants, we see that we may replace the number zero in condition 2) by any other number without essentially chang- ing the meaning of the condition. Consequently the following condition is satisfied

2') For each f e : ~ , each real a and each neighborhood _N' of ~, we can find a neighborhood N of ~ such that 2VC N' and f-LN e ~.

1.4 DEFINitION. - By a 5~-filter we mean a subset ~ of S such that:

a) If A e $ and A D B ~ C ~ , then A 6 C t .

b) If A and B e ~ , then A (~ B e d .

e) O~e2.

Annali di Matematica 3

18 HUGH GORDON: Compactifications Defined by Means, etc.

A maximal ~-f i l ter , ordering all JB-filters by inclusion, will be called a :~-ultrafilter. (Note that ~ f i l t e r ) > is actually an abbreviation for (<(~, 8)-filter>), since ~ as well as :~ enters into the definition).

1.5. - We omit the proofs of the following four lemmas, since the lem- mas may be proved in the order shown without any difficulty. Furthermore, the proofs are in all essentials identical with the corresponding proofs in [2; 2.5, 2.6, 2.13] and [1; Chap. 1, § 6, no. ~]. We do record explicit ly that ZoR:n' s Lemma is used.

LEM~[A. - Every :~-filter is contained in a :~-ultrafilter.

1.6 LEMMA. - Each non-empty zi e ~ belongs to a :~-ultrafilter.

1.7 L E ~ A . - Let ~ be a ~-ul t raf i l te r . If A ~ $ meets each B ~ , then A e ~ .

1.8 LEMMA - Let ~ be a :~-ultrafil ter. If A and B e S and A O B e ~ , then either A e ~ or B ~ .

1.9 LEMMA For each f e : ~ and each ~-u l t r a f i l t e r ~ , f (~ ) converges (as a filter base on the space R of real numbers).

P I t o o F . - ~or each A e ~ , f(C[)C[--Itf lI , l}/II], which is compact. Hence it suffices to show that f (~) has no more than one point of adherence. Sup~ pose on the contrary that ~ and ~ are distinct points of adherence of f(~). Let ~ = t ~ - - ~ i / 2 and, by 2') of 1.3, choose neighborhoods N~ and N2 of and ~ respectively such that N 1 C ( ~ - - e , a + e ) , N 2 C ( ~ - - ~ , ~ + e ) and f-1~1 and f-12¢2 both belong to $. f-~N1 and f-~/V2 are disjoint since _N~ and 2(~ are. Each A ~ ~ meets both f-~N1 and f-~2(2, since cIRf(A) con- tains both ~ and ~. Thus both of the disjoint sets f-~2V~ and f-~N2 belong to ~t by Lemma 1.7, which is absurd and completes the proof.

2. - The Construction of the Compaetifieation.

2.1 D:EFI~I~IO~. - Let X be the set of all ~-ul t raf i l ters . For each fs:~, we define f " .~:-->R by f ( ~ ) - - - l i m f(~I) for all 5 e 2 . Let :~ - - { f l f e : ~ } . We give f ( the weak topology determined by ~ ; in other words, we take the :~-cozero sets (i.e. the sets of tile form _ X \ f - l ( 0 ) w i t h f e J B ) a s a base for the topology of 2~.

2.2 RE~IAI~K - X need not be HAUSDOt~FF. For this reason, we introduce the space X later.

HUGH GORDON: Compactifications Defined by Means, etc. 19

2.3 LElgMA. - T h e mapping f--->~ is an isomorphism of the ring ~ onto the ring J~.

PROOS. - (f 4- g)-~ = lim (f + g)~t = lira f(~t) + lira g(~) = T(~t) + g(~); similarly for products. Suppose f @ 0 ; then for some w ~ X , f x@O. Choose by 2') of 1.3 a neighborhood N of fx such that f - ~ N ~ g and O~_~InN. Setting A ----- f - ~ we note that x ~ A and hence A is non-empty. Since also A belongs to $, there is a :~-ultrafil ter ~ containing A (Lemma 1.6). Since / '(~) -" lira f ( ~ ) ~ c l f(A) C c I N and 0 ~ cl N, f(~) ~ O. Thus f 4 0 implies f : # 0.

2.~ LEMMA. - Let f e J~ and c~ ~ X. Suppose f ( ~ ) ~ 0. Then there is an A e $ such that A ~ ~ but A ~ ~ [or each ~ ~)~ with f (~) _~-- 0.

PROOF. - Let ~ ---T(~); choose a neighborhood N of 0 in R such ~hat NC(--1~t/2, 1~1/2) and f - ~ N s ~ . Let A = f - 'N. Since A e ~ implies ~ =- lira f ( c~)ec l f (A)C[- - i~[ /2 , [~l/2], A~.~ . Suppose 7 ( ~ ) = 0 . Then for each B s ~ , 0 is a point of adherence of f (B)and hence f (B)meets ( - - [~[ /2 , [s~l/2 ). Thus each B e ~ meets A, and hence A s ~ by Lemma 1.7.

2.5 TIIEORE)I. .J~ is compact.

PROOF. - Let ~ be a collection of closed sets of X with the finite inter- section property. Consider the following property, which A e S may possess: There are C~, ..., C ~ e ~ such that each ~ e C ~ C~ ... (~ C~ contains A. Let $: be the set of all A e $ having this property.

W e claim

(Actually, it is easy to see, but irrelevant, that those sets are equal). Suppose ~ (h ~. Take C e ~ with ~ ~ C. Since C is closed and the topology of X is

the weak topology determined by ~ , we can find f e ~ such that f ( ~ ) ~ - 0 for all ~ E C but f ( c ~ ) ~ 0 . By the preceding lemm~, there is an A e S such that A ~ but A e ~ for each ~ with )7(~)_-0, hence for each ~ e C . Thus by the defining property of g, A e g. Since A ~ ~ , d tD ~" fails. This com. pletes the proof tilat t ~ e X I ~ D ~ } C N C .

To complete the proof of the theorem, we need only show that ~ is a JB-filter and apply Lemma 15. Suppose A~ and A 2 e ~ . Then there are C1, ..., C~, C,,+~, ..., C~ such that each ~ e C ~ ~ ... (h C,, contains A~ and each ~ E C,+~ (5 ... (~ C,~ contains A2. Hence each c~ ~ C~ N ... A C,~ contains both A~ and A2 and hence their intersection. Thus A~ N A2 e ~ . The rest is clear.

20 Hu~n GORDON: Compacti]ications Defined by Means, etc.

2.6. - Define an equivalence relation E on X by ~ is equivalent to ~ ' if [ ( 4 ) = f~ ' ) for all f e :~. Let f~=X/E and let ~ be the natural mapping. Certainly X is compact. For each f E ~ , we may use the equation fo ~ = f to define ?" X--> R. Surely each f is continuous and f--> f is an isomorphism of ~ onto : ~ - - i f l f e ~!. Since ~ separates points of l~, X is HAUSDOI~FF. By the STONE-WEIERSTRASS Theorem, ~ -- C(J~), the ring of continuous func- tions on X~ Consequently, and this summarizes our work, 3B is isomorphic to f~ = c(f().

2 . 7 . - For each x e X , { f ~ t f ( x ) = O ! is a maximal ideal of :[13; hence - - i f ~ :~If(~) - - 0 } is a maximal ideal of 3~. It is easy to show [2. Theo-

rem 4.6] that there is a u n i q u e p e X such that i q h - - i f e ~ 3 i f ( p ) - - O ! . Put i(x) -" p, thus defining a mapping i " X--> X. Since ~ separates points of X, i is injeetive. Identifying X with i(X) we have:

TKEORE~. - Let X and )~ be as above. There is a compact HAUSDORF:~ ~ space .7~, in which X is dense, such that each f ~ : ~ has a continuous exten- sion f to X. The totality :~ of such extensions is C(X).

PROOF. - It remains only to note that X is dense in X since if f van ishes throughout X, it vanishes throughout X also.

2.8 REMARK. - If we start with a completely regular space X and let be the ring of bounded continuous functions on X, then the :~-cozero sets form a base for the topology of X while ~-eozero-se ts form a base for that of X. Thus clearly the initial topology of X: is the relative topology induced on X by X.

3. - Convergence of ~ - u l t r a f l l t e r s .

3.1 LsM)~[A. - If A e S and pecl~:A, then there is a :~ ultrafi l ter which contains A and converges to p.

PROOF. - Let ~ consist of all sets in 8 which are intersections with A of neighborhoods of p in X. Clearly ~ is a ~ - f i t t e r (Definition 1.4) and A E ~ ; we show ~ converges to p. Let V be any neighborhood of p in X. Then there is an f e ~ such that fp = 1 but f q - - 0 for each q~ X \ V. By 2') of 1.3, we may choose a neighborhood AT of 1 such that N C(0, 2) and f - lNe $. Let B = A A f-llV. Note B ~ S, B -- A (~ f-iN and f - i N is a neigh. borhood of p in X. Thus B e ~ " by the definition of ~. Since 0 ~ N , ( X \ V ) N N f - i N - - 0 and thus f - l lVC V; it follows B C IT. Since V is an arbi trary neighborhood of p, ~ converges to p. It remains only to note that by Lemma 1.5 there is a ~ -u l t ra f i l t e r containing ~ and that this ~ ultrafi l ter is as required.

HUGH GORDON: Compacti]ications Defined by Means, etc. 21

3.2 TttEORE~I. - Each :~B ultrafi l ter converges to some /9 e )f. Each p e / ~ is the limit of some ~-ul t ra f i l te r .

PROOF. - Let ~t be a ~-ul t raf i l te r . By compactness, if ~ has at most one point of adherence, then ~ converges. For each point of adtlerence p of

and each f e ~ , ]~p is a point of adherence of f ( ~ ) . . B u t we know f (~) converges (Lemma 1.9). Hence }l) = lira f(~), i.e. }io is the same for all points of adherence p of ~I. Since :~ separates points of _~, ~. can have but one point of adherence.

Now let p e X be given. By 2) of 1.3 applied to f - - 0 , we see X eS . Applying the lemma with A : X, we have desired conclusion.

3.3 THEOREM. - In order that each point cf X be the limit of jus t one ~ -u l t r a f i l t e r it is necessary and sufficient that the following condition be satisfied: Whenever A~ and A2 are disjoint non-empty sets belonging to 8, then there is an f E ~ with f(A1) --{0} and f(A2)--{1}.

PROOF. - In view of the normali ty of 2~, the condition of the statement of the tl, eorem is equivalent to the following condition: Each pair of disjoint sets belonging to $ have disjoint closures in X.

Suppose this last condition holds and Cf~ and ~2 are distinct JB-ultra- filters. Then there is an A ~ e ~ , say, with A ~ 2 . Then for some A2~Cf2, A~ A A2 ~ .0 (Lemma 1.7). It follows from the condition that A~ and A2 have disjoint closures ia X. Since these closures contain the respective limits of ~ and ~2, these limits are distinct.

Towards proving the converse, suppose now A~ and A2 are disjoint sets belonging to $ and pecly~A~ A cl~A2. By lemma 3.1 we find ~ and Cf2 converging to p and containing A~ and A2 respectively. Surely ~ =4= ~ , and hence p is the limit of more than one ~-ul t raf i l te r .

3.4 REMARK. - Suppose X is completely regular and ~ is the ring of bounded continuous functions on X. If 8 is the set of all zero-sets, the con. dition of the theorem is always satisfied. If $ is the set of all cozero-sets, the condition is satisfied precisely when X is an F space [2, 14 N4].

4. - Real and Hyperrea l Points .

4.1 D~FrN~IO~S. - We call p e X real if f/9 ----- 0 implies f0c = 0 for some x e X. Otherwise we call p hyperreal.

4.2 REMARK. - Thus the real points are the ones corresponding to the weak maximal positive cones of LoRcE [6], while the hyperreal points corre- spond to the strong cones. Additional propert ies of this distinction among the points of 2~ will be found in the paper jus t cited.

22 HUGH GORDON: Compactifications Defined by Means, etc.

4 3 [JEIVIMA.- Let p be real. Let Z ~ , Z 2 , ... be :~-zero-sets with p e c l ~ Z n c o

for all n. Then p e c l ~ C1 Z~.

PROOF. - Suppos not.e Then we find g ~ : ~ such that g p = 0 but g q - - 1 for all q e c l ("1 Z~. Choose f~, f~, ... e • such that fn~(0)= Z . ; note ] ~ p - 0. Pu t :

oo

h = + (I f . l A

Then h p - - 0 ; hence h x = O for some ~ e X . It follows that g x - ' O and hence .~ ~ C~ Z~. But it also follows for each n that f ~ = 0 and hence x e Z~. This contradiction completes the proof.

4.4 RE]~iARK.- If p is hyperreal, it may happen that p belongs to the closure of each of two disjoint :~-zero-sets as pointed out in the introduction.

4.5 TI~EORE~I. Let p eT~ and let ~ be a fil~er b~se on X converging to p (as a filter base on 7f). Then the following conditions are equivalent :

a) p is real.

b) Whenever Z~, Z2, ... are o o

set belonging to ~ , then N Z,~ ~ 0 .

~-zero-se ts each of them containing a

c) Whenever C~, C2, ... are ~-cozero-se ts none of them containing a c o

set belonging to ~, then U C~ ::~ X.

PROOF. - C) ~ b): Let Z~, Z2, . . . be :~-zero-sets each containing a set belonging to ~. Put C~ ~ X \ Z ~ for each n and observe no C~ can contain a set belonging to ~. Thus (2 C~ ~ X and, taking complements, (5 Z, ~ 0 ,

a) ~ c): Let C1, C2,... be ~-cozero-se ts none of them containing a set belonging to ~. For each n, let Z . be the ~ cozero-set X \ C ~ . Lei U

be any neighborhood of p in 7~. Then, since ~ converges to p, there is a set in ~ contained in U; hence this set in ~ is contained in U N X, since each set in ~ is contained in X. Thus for no u can we have C ~ U N X since by hypothesis no C, contains a set in ~. It follows that each Z~ meets each neighborhood U of p ; in short, p e c l ~ Z ~ for all n. Then, by the lemma, p E c l ( ~ Z ~ and hence C ~ Z ~ 0 . Taking complements, U C ~ X .

b) ~ a): Suppose that a) fails, i.e. that p is hyperreal. Then we can find f s Ji3 such that ~o = 0 but ]'x =~ 0 for each x e X . For each positive in- teger n, put Q, = { q ~ Jf t l fql < 1 /n I and Z , -- Q,, A X. Then each Q~ is a

HUGH GORDON: Compactifications Defined by Means, etc. 23

neighborhood of p in X, and hence, since ~ converges to p, contains a set belonging to ~. Since every set in ~ is contained in X, we see that each Zn contains a set belonging to ~. Since clearly x E (5 Zn implies fn~--0, we have N Zn = 0. Thus b) fails, which completes the prooL

4.6. - I t is of some interest to apply the preceding theorem to ~-ul t ra- filters when 8 is either the 3~-zero-sets or the ~-eozero-sets. ]f 8 consists of all :~-cozero-sets, we obtain the following theorem from the equi~alence of a) and c) of the preceding theorem:

T H E O R E ~ . - L e t p e ~ : and let ~ be a :~ ultrafilter, with respect to the set S of all ~-cozero-sets. Suppose ~ converges to p. Then in order that p be real it is necessary and sufficient that every countable covering of X by ~-cozero-sets contain a set in ~ .

4 . 7 . - As for the case where S consists of the ~-zero sets, the situa- tion is best described by means of two contrasting theorems:

THEOREM. - Let p be real and let S consist of all the 5~-zero-sets. Then "- I A e $ IP e c l ~A } is the only 5~-ul~rafilter converging to p. This :~-ultra-

filter is closed under countable intersection.

PRooF. - Lemma 4.3 states that ~ is closed under countable intersection, hence surely finite intersection. It follows at once that ~ is a 5~-filter. But every :~-filter having p for a point of adherence is contained in ~ . Thus every ~-u l t ra f i l t e r converging to p must be contained in, and hence equal to, ~. By Theorem 3.2, there is such a ~ ultrafilter.

4.8 THEOREM. - Let p be hyperreal and let ~ consist of all the 5~-zoro- sets. Then no ~ ul traf i l ter converging to 19 has the countable intersection property.

PROOF. - We use the implication b) ~ a) of Theorem 4.5.

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1960. [3] E. HE~VITT~ Rings of real-valued conti~uous functions, I, Trans. Amer. Math. Soc. 6=~

(1948}, pp. 45.99. [4] C.W. KOHLS, Ideals in rings of continuous fu~ctio~*s, Fund. Maih. 4-5 (1957), pp. 28.50. [5] E.R. LOReH, L'inidgration daus les espaces gdndraux~ Bull. See. Math. France 88

(1960), pp. 469.497. [6] - - - - , Compactification, Baire f~ctions, a~ld Daniell integratio~b Acta Set. Math.

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