TOTALLY DISCONNECTED COMPACTIFICATIONS

There is a one-to-one order preserving correspondence between totally disconnected compactifications of a topological space and certain Boolean algebras of open closed subsets on it.


INTRODUCTION.
One of the interesting consequences of Gelfand theory is the correspondence between compactification of a completely regular space S and certain subalgebras of the algebra C(S) of all bounded continuous complex-valued functions on it. This correspondence is described in 43 of Gel'farad et al. [1]. Theorem 1 in 43 of Gel'fand et al. [1] states that each Hausdorff compa,'tification of S corresponds to a subalgebra A of C(S) which is closed under involution and is regular in the sense that members of A separate points from closed subsets.. In this paper we shall show that there is a similar consequence of Stone theory about Boolean rings. We shall prove that there is one-to-one order preserving correspondence between certain compactifications of a totally disconnected space S and certain Boolean rings of open-closed subsets of the space S. (In fact, we would be justified in referring to those Boolean rings as regular Boolean rings (members of these rings also separate points and closed sets)). There is an interesting analogy between consequences of Gelfand and Stone's theories. 2. MAIN RESULT.
Let S be a totally disconnected space (see Simmons [6, p 149]). This means that for any s, E S there is an open closed subset of S containing s but excluding (Note that S is a Hausdorf space). (Note also that our definition differs somewhat from definitions given in Johnstone [2], Koppelberg [4] and Monk and Bonnet [5].) Let :(S) be the space of all open closed subsets of S. Then L:(S) is a Boolean algebra (Simmons [6, p 345]) under standard set theoretic operations (as well as a Boolean ring under certain algebraic operations (symmetric differences and intersections)).
The main result deals with those compactifications of S that are totally disconnected. Ac-cording to Kelley [3, p 151] a compactification of S is a compact space Q D S such that S is dense in Q and the topology of S is the (relative) topology induced by Q. THEOREM 1. For each totally disconnected compactification Q of S there exists a Boolean subring L(Q) of (S) with the following properties: (1) L(Q) is isomorphic to the Boolean ring : ( (2) there exists a totally disconnected compactification Q(L) such that L is isomorphic to (Q(L)).
PROOF. Let Q be a totally disconnected compactification of S. Consider the family Then L(Q)is a Boolean algebra isomorphic to (Q) (under the correspondence a N S a)(the fact that the correspondence "G f3,5' G" is one-to-one follows from the fact that each G (Q) is both open and closed, and S is dense). Using Theorem C on page 150 of Simmons [6], it is easy to verify that L(Q) satisfies property (2). Now let L be any Boolean subalgebra of (S) satisfying (2). Let Q(L) be the set of all non-zero homomorphisms q of L into Z2 {0, 1} (the smallest Boolean ring (which is also a field)). Then Q is totally disconnected compact space with respect to the topology whose base B can be described as follows (Simmons  (where e is the characteristic function of G" e(s) 1 if s e G and a(s) 0 if s G). Now note that the map " s q, is one-to-one (because of property (2)) and continuous (since -l(Ua) G). Property (2) implies that maps open sets into open sets (every open set in S is a union of members G of L and (G) Ua t3 (S)).
To show that 5' (or rather (S)) is dense in Q, consider any q0 Q and any open subset O of Q with q0 Q. Take G E L, G :/: , such that q0 Ue C O. Then Ue contains plenty of members of the form q with s q S. This means that S is dense in Q.
Thus Q is a compactification of S. It is not difficult to show that L is isomorphic to (Q) (in fact, this is the consequence of Stone Representation Theorem on p. 351 in Simmons [6]). Now let " be the class of all totally disconnected compactifications of S and let E be the class of all Boolean subalgebras of (S) satisfying condition (2) of Theorem 1 above. Define the partial order "<" on " as it is done on page 151 of Kelley [3]: if Q1, Q2 then Q < Q2 if there exists a continuous mapping T Q2 Q such that T, s for all s S. Order E by inclusion: if L, L2 E (S), then L1 _< L2 if L1 is a Boolean subalgebra of L2. THEOREM 2. The mappings Q L(Q) and L Q(L) considered in Theorem 1 above are one-to-one and preserve partial orders of 2" and .
PROOF. First ssertion follows fro,n the facts that Q is homeomorphic to Q(L(Q)) (follows from Theorem 2, p 380 of Stone [7]) and that L is isomorphic to L(Q(L)). (Stone Representation Theorem on page 351 of Simmons [6]).
Let us show that the correspondence L Q(L) preserves the partial order. Let T Q2 Q be a continuous map of a compactification Q2 into a eompactification Q1 (Q, Q2 6 .T, Q <_ Q2) such that T, s for all s 6 S (for simplicity we identify members s of 5' with their corresponding members qs in Qa and Q2). (2.9) Then T(Va) C Ua and so T is continuous on q0. It is also easy to see that T, s for each s 6 S.
3. AN EXAMPLE. It is clear that total disconnectedness of the space S above does not by itself guarantee existence of Boolean algebras of open closed subsets of S that satisfy condition (2) (in Theorem 1) above. Because of this we give an example of a space which has p!enty of Boolean algebras with property (2).
In this paper the term "open interval" means any set of the form (a, b) {x: a < x < b}, The author feels that it is appropriate to make the following remark concerning completely regular spaces mentioned at the beginning of the paper. As it was already stated above, there is a one-to-one correspondence Q B between Hausdorff compactifications Q of a completely regular spae S and Banach subalgebras B of C(S) which are closed under conjugation and which separate points and closed sets in S. It turns out that this correspondence preserves also the partial order. In other words, if QI and Q2 are two Hausdorff compactifications of S such that Q1 < Q2 (as defined above) and B1, B2 are corresponding regular Banach algebras, then B1 C_ B. Proof of this fact is a straightforward modification of the proof of Theorem 2 above.