REALCOMPACTIFICATION AND REPLETENESS OF WALLMAN SPACES

The extension of bounded lattice continuous functions on an arbitrary set X to the set of lattice regular zero-one measures on an algebra generated by a lattice (a Wallman-type space) is investigated.

by L, and M(L) those bounded finitely additive measures on A(L ), and m t(L those elements of M(L) which are L-regular while M n(L) denotes those elements of M n(#) which are countablv additive.The zero-one valued members of the above are designated by I(L ), n(L ), and respectively.For A A(L ), w(A) {u 1I(L_)[ u(A) 1}, w(L {u'(L)IL L ), then ln(L with the topology of closed sets ru'(L) of arbitrary intersections of sets of w(L) is a compact, T topological space.It is one of the Wallman type spaces.Assuming L is disjunctive then it is T if and only if L is normal.
We begin by considering briefly, because of their importance, certain fundamental properties of normal lattices.Then we proceed to a consideration of ln(L ), and the extension of bounded lattices continuous functions on x to II(L ).These results are generally knovn (see [8]) but we give somewhat shorter more direct proofs here.
We next consider the space Q(L) of measures in t(L which integrate al___[1 lattice continuous functions on X, and show its relationship to t(L ), and under suitable conditions, its relationship to the G-closure of X in In(L ).
Finally, we consider the Wallman type space ln(_L ), and the lattice w(_L ), where for A A(_L ), w,(A) {u IRa(L_)[ u(A) 1}, and where w,(_L {w(L)[l, _L }.It is well-known that if _L is disj'unctive then w(_L) is replete.We consider in this space the lattice of closed sets rw(L and its associated lattice of zero sets, and investigate their repleteness thus obtaining sufficient conditions for the space 1,q*(_L to be realcompact. Our notations and terminology is consistent with [1, 3, 5, 6, 11].However, the main definitions and notations used throughout the paper are presented for the reader's convenience in section 2(a).We note also that a number of results on normal lattices in section 2(b) are related to work of [4 9].Let x be an abstract set, and __L the lattice of subsets of x.We assume that , X _L for most of our results.First: Lattice Terminology: A(_L is the algebra generated by _L.a(_L) is the a-algebra generated by _L.6(_L is the lattice of all countable intersections of sets from _L. _L is a delta lattice (6-lattice) if r(_L is the lattice of arbitrary intersections of sets of _L. _L is complemented if L _L > L' _L (prime denotes complement), that is, _L is an algebra.
_L is separating, if for any two elements z y of X, there exists an element L _L such that and y L.
_L is T if, for any two elements z # y of X, there exists A,B L_. such that X A' and u B' and _L is disjunctive if for any z X and A _L such that r A, there exists a B L such that B and At3B=.
_L is regular if for any z X, and A _L such that z A there exist B, C _.L such that A C C' and B' t3 C' .
_L is normal if for all L a, L2 _L.such that L: t3 L there exists /q, L _L such that L C LI', L C L2', and 1' CI 2' $" __L is compact if every covering of X by elements of __L' has a finite subcovering.
_L is countably compact if every countable covering of x by.elements of _L' "has a finite subcovering.
_L is LindelSf if every covering of x by elements of _L' has a countable subcovering.
__L is countably paracompact if whenever A. ,A._L there exists B. _L such that A. C B.' and B,' .
L is complement generated if, for L L there exists L, L such that L n=l n" It is well known that if _L is complement generated then _L is countably paracompact.
We will use the following notations.

M(L
the set of a-smooth measures on/' of M(L ).
M(L the set of a-smooth measures on A(/' of M(L ).
Let C(/' be the set of all real-valued _L-continuous functions defined on x, where I:X--,R is called /,-continuous if I-I(E)6/' for any closed set E C R. If X is a topological space, C(X) denotes the continuous functions on x or equivalently we can write C(X)= C(F) where F is the lattice of closed sets of x. z(/' is the lattice of zero sets of functions in C(/' ).
We have for A, B fi A(/' ): (1) w(A U B) w(A)Uw(B) (2) w(ACIB) w(A)f3w(B) (3) w(A)' w(A') (4) w(A(L )) A(w(L )) (5 Note w(_L is a lattice and if L is disjunctive then w(A)= w(B) if and only if A B. The Wallman topology is obtained by taking w(_.L as a base for the closed sets of a topology on l a(_L ).< I(_L ), rw(_L)> is the general Wallman space associated with x and __L.Note we have w(L) Z for L _L if _L is separating and disjunctive.We also define: w,,(A) {u Irt"(L_.u(A) 1} where A tE A(/' ), and note w(_L n In(L w(L ).
We now consider two lattices.Let __L and _L denote lattices of subsets, of X where _.L C L _L1 semi-separates _L2 if AL1, BL2 and AnB= implies there exists C_L_I, BCC and A n C ._L separates _L if A, B __L and A n B implies there exists C, DtE L such that A C C,B C D, and C n D ./' is _L rcountable paracompact if for every sequences {B,,} of sets of _L 2, such that B,, there exists {A, q _L 1} such that A,' and B, C A,,'.
_L is Lcb if given B,, , B. _L there exists {An},A .L such that A,, and B C A n.
If v e M(L2) then by v A() we xnean v restricted to A(_L1).We state the following well known results: Let L C _L2 be two lattices of subsets of X.If __L semiseparates _L2 then for v e M R(_L), u=vlA(L) MR(L 1).
Suppose L C L are two lattices of subsets of X.Then if tt E M(_L 1), u extends to t, E Mn(_L ).Moreover, the extension is unique if L separates L 2.
We will frequently assume in the sequel that _LC_L2 and _L2 is _L countably paracompact or countably bounded, but we note that this is unnecessary in certain situations as the following facts listed below show: (1) If L is _L1 countably bounded and if _L is countably paracompact (e.g., if _L1 is complement generated) then _L is _L countably paracompact.
(3) Suppose L is L countably paracompact and _L semiseparates _L then L is L countably bounded.

2.'(b)NORMAL LATTICES AND MEASURES.
In this section we will consider a number of measure implications of normal lattices and other special lattices as well as converse implications.We first note: THEOREM 2.1.Let _L be a complemented generated lattice.The tt I(_L') implies PROOF.Since _L is complemented generated then L is countably paracompact and therefore I(_L')c I(L_ ).Therefore it suffices to show u fi l n(L ), but this is easy for if L _L then L= ,=ILn"L" _L all n, and we may assume that the L,' .Now if u(L)=O, and if all u(L,') then ,,=t31 nL'= and u(L,,'taL')= all n which is a contradiction since u I(L').It follows that u(L) i,ff{u(Z') L C ,', q _L and this implies u E IR(_L ).
REMARK.This theorem is equivalent to the following: Let L be normal and let.v _< u(_L) where v I(_L and u I(L ).Then u(L') sup{v(,) , C L',. L }. Next we show that actually the property in Theorem 2.4 or equivalently the one in the remark characterizes normal lattices, i.e., THEOREM 2.5.Suppose u la(_L and p _< u(/, where p I(/,') and u(L') 1, L /, implies L'D A _L such that p(A)= 1.Then _L is normal.
Next we consider a pair of lattices _L , _ 2 of X such that/, c/, 2, then we have: THEOREM 2.8.If _L separates/, then _L is normal if and only if/, 2 is normal.

THE WALLMAN SPACE I(_L).
We give here a brief discussion of the general Wallman space (see also [11]).Consider the set I a(_L and the lattice of subsets w(_L ).It is well-known that w(_L is compact and it is not difficult to show: (c) w(_L)is r:.Now since w(_L) is compact, rw(L_) the topology of closed sets, is compact and w(L) separates rw(L_ ), and by Theorem 2.8 w(_L) is normal if and only if rw(L_) is normal.< In(L_ ), ru,(L )> is a compact t.opological space and it is always T 1. Assuming _L is disjunctive, it is T if and only if _L is normal.Next, let _L be a 8-normal lattice of subsets of X, then the Alexandroff representation theorem (see [1]) yields for the conjugate space of Cb(_L ), namely Cb(_L )'= MR(_L where to any E C(_L )" there corresponds a unique u E MR(_L such that ,(f) f fdu, for all f C(L ).
x A net {u,} in M R(_L converges to u in MR(L in the weak topology if and only if f fdua--, f fdu for all f C(_L ).We shall denote weak convergence by w*.
We assume now that _L is -normal, separating and disjunctive.Let f Cb(L we define 7 on IR(L by (u) f fdu where u e IR(_L ).X THEOREM 3.6.7 e CUR(L_ )).w PROOF.Let u u o.We must show that (u)-(Uo) which means u,-u 0. For u o w(L') we w have u w(L') for all a > a o as u, u o.Therefore, u(L')= 1, a > a0, which implies lim u(L')= 1.
OR .
PROOF.By Theorems 4.1 and 4.2 and the trivial observation that In'(/.)cI(I,)t')l,,(L_'), the result is proved.Following Varadarajan who considered the lattice of zero sets in a Tychonoff space, we introduce DEFINITION.The Sequence {B,} in __L is called regular if B,.," and there exists A, in _L such that B. c An'C Br, + for all n.THEOREM 4.5.Let X be an abstract set and L a b-normal lattice of subsets which is also countably paracompact.Let {A,} in _L, A, .Then there exists a regular sequence {C,} such that C c A n" for all n.
PROOF.Since A and since L is countably paracompact then there exists {Bn} in _L with ARC Bn" 4. Now we show by induction that for any n we have {CK},{DK} in _L with AKCCK'CDKC(Bh.'VC'K_I)where K=I follows by normality.
We have the following application: For _L 6-normal, z(_L C _L where z(_L consists of all sets of L of the form L Ln', L L for all n, (see [1]).Now z(_.L separates L and z(_L is normal and n=l countably paracompact.Therefore by Theorem 4.7 we have Ina(z(L_))=Q(z(L_.)).Now using Theorem 4.11 we have Q(_L IR(L C Ia(_L").Also if Ia(L" C Ia(L then Q(_L Ina(_L by Theorem 4.8. REMARK.We recall that if x is Tychonoff space and if _L z, the lattice of zero sets then (IRa(z), rwa(z)) is the realcompactification u(x) of x.Now we consider other criterion for Q(L)= In(L_)CIa(L").If X is a topological space and if A C X we denote by ]6 the GT-closure of A. Now if X is an abstract set and L as usual is a separating disjunctive b-normal lattice of subsets then we can view X embedded in Q(_L); we have x c Q(_L c ln(_L ).In fact, using Theorem 4.3 we have X c In'(_L C IR(_L t3 I,(L ") C Q(_L C IR(L ).THEOREM 4.12.R$ c Q(_L where X " is the G-closure of X in the Wallman space In(_L ).
PROOF.If Q(_L)= and if uQ(L_) then u I,(_L') by the previous theorem.While if Q(L)CI,(_L') then we must have Q(_L)c for if not then there exists GG such that u G c IIt(_L X where u IR(__L ).Therefore u ,,=t3 O c Ir(_L where On is an open set, which implies u O for all n.Now w(Ln" is an open set for L __L, therefore u w(L,')C: O, which yields u f3= iw(L C ,= 0 Therefore there exists u Q(_L) such that u I " 1 = lW(Ln') where the w(Ln" and where L, L and w(L,() C la(L )-X, but then u(zn" for all n and L n" which is a ctntradiction.Thus Q(_L) c and then by Theorem 4.12, Q(_) X Using the previous theorem and Theorem 4.2 we have: REMARK.We note that Q(_)= l rt(L_ if and only if Ct,(L)= C(L); this situation arises in particular if C(_L) consists only of constant functions.(see below) 5. THE WALLMAN SPACE l(_L).