ON CERTAIN WALLMAN SPACES

Several generalized Wallman type spaces are considered as well as various lattices of subsets therein. In particular, regularity of these lattices and consequences are investigated. Also considered are necessary and sufficient conditions for these lattices to be Lindelof as well as replete, prime complete, and fully replete.


INTRODUCTION.
Let X be an arbitrary set and a lattice of subsets of X such that ,X E Z..4() designate, the algebra generated by Z, and I() is the set of zero-one valued finitely additive meastLres on I,() denotes those elements of I(L;) that are a-smooth on , and I'(/) those which are a-smooth on .4(L;)i.e. are countably additive.IR(/) denotes those elements of I(/) which are L;-regular; while I(/;) denotes those elements of IR(L;) which are also in Is(L;), and consequently in I().These various sets of measures have distinguishing lattices of subsets within them which are taken as bases for closed set topologies, and are called generalized Wallman spaces.Some of these spaces such as la() and I() where is disjunctive are well known and special cases give known topological spaces.
We wish to investigate in some detail the spaces I(L:) and 1"(.) and the lattices of these space (see below for explicit definitions).In particular, we investigate when.these lattices are regular an,l subsequent consequences of regularity.We also fred necessary and sufficient conditiorLs for spaces to be Lindelhf and also, for a number of these lattices, consider questions of repletene-.-prime completeness and fully repleteness.
We begin in section 2 with some background information on these spaces and lattce.summarize a variety of known results.Then in section 3 and 4 we pursue our investigat()n indicated above.
The author takes pleasure in acknowledging his indebtedness to the referee for greatly impr()vu; the presentation of this paper.

BACKGROUND AND NOTATIONS.
We follow standard notation and terminology such as appears in [1, 2,4,6,11].We review some of this here for the reader's convenience.
Let X be a set and a lattice of subsets of X such that , X belong to/:.A() denotes the algebra generated by L:, and M(.) denotes those finitely additive, bounded, non-negative measures on A(L:) which are non-trivial.Ma(L:) denotes these elements of M() which are a-smooth on That is, p e Ma(), if for L. e and L. {, we have #(L,) O. M"(.) denotes these elements of M(/:) which are a-smooth on.'4() or equivalently here, are countably additive.
Next, Mn(E) denotes these p e M() which axe E-regulax.That is p(A)= sup{p(L)l L C A,L E} where A A(E). Note, ifp 6-Mn(E) and p Ma(.) then / e M(:) Such measures are denoted by M(E).Finally the zero-one valued subsets of the above axe denoted by I(E).Ia(E),Ia(E),In(E) and I,(:) respectively.
If p M(E), the support of p,S(p) is defined as follows.S(p) N{L E[p(L) p(X)}.We recall (see [2,6,9]).E is compact if and only if S() y(: l} for all, I(E), or equivalently S() # 8 for all p In(E), or equivalently S(p) # ) for all II(E).Here, r II() if r" E {0, }, r(X) 1, and r is monotone, and r(A '1 B) r(A)r(B),A,B , and r($) 0. Ha(E) denotes these r H(E) which are a-smooth on E. For r II(),S(r) is defined as for p I(E).With these definitions in place we have: is countably compact (c.c.) if and only if I() Ia(E), equivalently In(E) I(E), or H(E) Ha(/:). is aindelhf if and only if for all r Ha(E), S(r) # q}.: is replete if S(p) # } for all p I(E).E is prime complete if S(p) # # for all p Ia(E), and is fully replete if S(p) # {} for all p We can characterize other more familiar lattice-topological concepts in terms of these measures For example (see [2,6,9]) E is disjunctive if and only if pz In(C) for all x X, where p is the Dirac measure concentrated at z. E is regular if and only if for pl,p2 I(E) with pl _< 2(:), that i ,() < ,,() o u e s(,,) s(,,).E is T if and only if S(p)   or a singleton for all I(E), and E is normal if and only if for / _< vl(E),p <_ v(E) where/ I(E),v,v In(E), we have vl v. Finally, we note that various sets of zero-one valued measures on A(E) can be topologized using Wallmaa topologies.Let I be any subset of I(E), e.g.In(E),Ia(E), etc.Let A A(E), and denote by H(A)= {p I[p(A)= 1}.Then for A,B A(.) we have: H(ADB) H(A)kJH(B),H(AcqB)= H(A) It(B),A D B implies//(A) D It(B).If {U z e X} C I. Then H(A) ::) H(B) implies A D B. Also, I-H(A) H(A') where throughout the prime will designate the complementaxy set.The Wallman topology on I is obtained by taking the set of all H(L), E as a ba.
for the closed sets.In the particular cases of I I(E),In(E),Ia(E),I(E),I(E) denote these bases by V(E), W(E)Va(E), Va(), Wa(E) respectively.Some of these spaces have been thorouglfly investigated.In particular In(C), rW(E) (rE in general denotes the lattice of arbitrary intersection., of sets of ) which is a compact T space, and assuming is disjunctive then it is T if a,l td.v if E is normal.I(E), rV(E) is compact To.If E is disjunctive then the lattice Wa(E) in I(   replete.We propose to investigate further topological properties of these general Wallma, and to relate some of these topological properties to the underlying lattice E.
Related material can be found in [4,7,10].A few of these results will be cited in the next to show how it interrelates with our work. 3. TOPOLOGICAL PROPERTIES OF THE WALLMAN SPACES.
We note first some known facts about various Wallman spaces, (see [2,9,11,12]).W(E) is a compact lattice in In(t:), V(E) is a compact lattice in I().If E is disjunctive, then Wa(E) is replete in I.,().Va(E) is prime complete in I,(E), and V'(E) is fully replete in Ia(E).Moreover, we have (see [8]).THEOREM 3.1.I(), Y() i compact To.Alo, V() i TI if and only if I() I(E), which i equivalent to V() being disjunctive and i abo equivalent to V() being regular.
REMARK.I() In(E) is equivalent to being an algebra.Various proof of this appear in [8,9].It is even true for abstract'distributive lattice (see [3]).
REMARK: Since W() is compact, rW() is compact and IR(L), rW() gives well-known compactifications of X in the case where X is a topological space, and consist of closed sets, zero sets, clopen sets, etc.
We consider now W,() in I().If is disjunctive, then W,() is replete.It is prime complete if and only if (3.1)For any # I() there exist a v I() such that # < u().See [10].Also if we consider I() with the topology of closed sets rW,() then, if is disjunctive, I(), rW() is Lindelhf if and only if (3.2) For any 7r E II() there exist a p E I() such that 7r _< p().
In a similar maxmer, considering the pair I() and V,(), we have that the topological space I(), r V, () is Lindelhf if and only if (3.3)For any r II,() there exist a # I,() such that 7r < #().Again see [10] for details.We further note (details can be found in [5]).THEOREM 3.3.If i diajunctine, then in I() the lattice W() is fully replete :f and only if (3.4)For any p E I() there exists a v I(g) such that # _< v().We now consider the pair I () and Y ().
Then we have 1) If (3.5) holds and if is fully replete, then/: is Lindelhf.
It has been shown in [7] that if is disjunctive, then W() is regular if and only if the following condition holds.