Bases of Countable Boolean AlgebrasAuthor(s): R. S. PierceSource: The Journal of Symbolic Logic, Vol. 38, No. 2 (Jun., 1973), pp. 212-214Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2272057 .

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THE JOURNAL OF SYMBOLIC LOGIC

Volume 38, Number 2, June 1973

BASES OF COUNTABLE BOOLEAN ALGEBRAS

R. S. PIERCE1

The purpose of this note is to give a short proof of a conjecture of Feiner that every countable Boolean algebra has an ordered basis that is a lexicographic sum of well-ordered sets over the ordered set -y of all rational numbers. Actually, we prove a slightly more precise fact, which is formulated below as Theorem 3. An earlier proof of Feiner's conjecture was obtained by David Cossack (unpublished), using a different method.

Our proof will use the following property2 of Cantor's dyadic discontinuum D. LEMMA 1. Let B be a dense subset of D, and suppose that X is a countable set of

points in D. Then there is a homeomorphism of D onto itself that maps X into B. PROOF. Let X = {X1, X2, X3, * *}. Being homeomorphic to a countable product

of two point spaces, D is a complete metric space with the property that each of its points has arbitrarily small, metrically homogeneous, compact, open neighbor- hoods. Thus, it is possible to construct a sequence (po (P, T2, . . * of isometrics of D such that 9p is the identity mapping on D; for every x E D, the distance from x to Tk(X) is at most 2 k; if j < k, then Tk(Tk-1 *. TOX) = (k- I x oxj; and (Pk((Pk 1 .PoXk) e B. These properties, together with the completeness of D, guarantee that limk. Tk(pk -I... po exists, and it is easy to check that this limit is an isometry of D that carries X into B.

Our proof of Feiner's conjecture is based on a lemma that is a variant of Theorem 1.2 in [2]. The proof of this lemma employs a construction given by Reichbach in [3], and it is considerably shorter than the original proof of 1.2 in [2].

Let I denote the unit interval [0, 1]. The Cantor set will be written in the form D = I - Un<(JJnf where Jn = (an, bn), and these intervals are disjoint. Denote Y = {bn: n < w}. The definition of D is such that Y is dense in D.

If T is any Hausdorff space, and e is any ordinal number, then the expression T(? stands for the transfinite topological derivative of order e.

LEMMA 2. Let V be a space that is homeomorphic to D, and suppose that V- V0 D V1 D * * * D V, D * * *, e < i, where each V! is closed and nonempty, and it is a countable ordinal. Then there is a closed subset Z of I with D c Z, and a homeomor- phism cp of V onto D such that

(a) Z(Ut) = D and Z - D is homeomorphic to wa;

Received October 12, 1971. 1 This research was supported in part by the National Science Foundation grant GP-29248.

The author is grateful to Professor W. P. Hanf for pointing out Feiner's conjecture, and later mentioning Cossack's proof of it.

2 According to Professor R. D. Anderson, this lemma is well-known folklore of point set topology. However, the author was not able to find it stated explicitly in the literature. For this reason, a brief proof of the result is included in the paper.

212 ( 1973, Association for Symbolic Logic

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BASES OF COUNTABLE BOOLEAN ALGEBRAS 213

(b) (Z'' - D)- n D = p(V~) for all t < Mt;

(c) K& = Z rn [a,, bJ) is order isomorphic to W@(n, for some countable ordinal ae(n);

(d) the points of Z that are covered (in the ordering of Z inherited from the usual ordering of I) are exactly the elements of Un<coK

PROOF. Since tk is countable, there is a countable, dense subset W of V such that W: = W n V: is dense in V: for each e < tk. For convenience, define WA = 0. By Lemma 1, there is a homeomorphism p of V onto D such that T(W) c Y. For n < w, define ae(n) = min{f < ,t: bn 0 T(W,)}. Note that bn e T(WO) if and only if f < a(n). Define sets Kn C [an, bn) satisfying

(1) Kn is relatively closed in [an, bn); (2) Kn is order isomorphic to W,(n'; (3) an Kn; (4) bn is a limit point of Kn. Such sets exist because ae(n) is countable. By Lemma 4.5 in [2], we have (5) Kn(? = 0 if e > a(n); (6) bn is a limit point of Kn?( if e < ae(n).

Finally, define (7) Z = D U Un<wKn

Since I - Z = Un<,(Jn - Kn), and each Jn - Kn is a nonempty union of open intervals by (1)-(4), it follows that Z is closed in I. The assertion (c) of the lemma is clear from the definition of Z, and (d) follows easily from (2) and (3). Since Kn - {an} is open in Z, and Z4 =-D(4) = D for all e, we have Z(4) 0 Kn = Kn? U {an}, and Z(4 = Z(4) (D U Un<wKn) = D u Un<(oKnM. Thus, by (5),

(8) Za = D U U{Kn(4: ae(n) > e}. In particular, Z(e) = D. If bn e Tp( Wi), then t < ae(n) so that, by (6), bn (Z( - D) .

Therefore, T(V,) = (W:)- c (Z4 - D)r n D. To reverse this inclusion, sup- pose that x E D - i(V,). Then there is a neighborhood N of x such that N n

q((W4) = 0. We can assume that N = (br, as) for suitable r and s. It follows that if n < w is such that Jn n N =# 0, then bn e N, and, consequently, bn 0 p((W4). Therefore, ae(n) < e so that, by (8), N n (Z)- D) = 0. Thus, x 0 (Z a - D) -, which establishes (b). To prove the last part of (a), note that (Z - D)4 = Z4'- D is not compact if e < tk, and (Z - D)(") = 0. By Proposition 3.2 in [2], Z - D is homeomorphic to wt.

We can now prove Feiner's conjecture. THEOREM 3. Let A be a countable Boolean algebra. Then A has an ordered basis

of one of the forms wA n + 1, with A a countable ordinal and n < w; , 6,War + 1, with -q the countable dense order type and each ac. a countable ordinal; or a sum of these two types.

PROOF. Let X be the Boolean space associated with A. By the discussion in ?3 of [2], X can be written as a disjoint union X = U1 U U2, with U1 and U2 closed in X, and

(1) U1 = 0, or U1 is homeomorphic to wA n + 1, A countable, n < w;

(2) U2 = 0, or U2 ) is homeomorphic to D and U2 - U) is homeomorphic to wi, tk countable.

This topological decomposition corresponds to a direct product A = B1 x B2,

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214 R. S. PIERCE

where U1 is the Boolean space of B1 and U2 is the Boolean space of B2. By Theorem 2.5 of [1], B1 has an ordered basis of type wA .n + 1. So consider U2, which we may assume to be nonempty. Let V = U(,) and, for 6 < /i, let V: = (U24'- V)- n V. By Lemma 2, we can construct Z c I, and a homeomorphism 9 of V onto D such that Z(u) = D, Z - Z(#) is homeomorphic to co, and T(V~) = (Z(4)-D)- n D for all e < [t. By Theorem 1.1 in [2], p extends to a homeomorphism of U2 onto Z. Our proof is completed by the observation that {an: ni < w} has order type -y, so that, by (c) and (d) of Lemma 2 and Theorem 2.5 of [1], B2 has an ordered basis of type 2;r w`ar + 1, where each a, is countable.

REFERENCES

[1] R. D. MAYER and R. S. PIERCE, Boolean algebras with ordered bases, Pacific Journal of Mathematics, vol. 10 (1960), pp. 925-942.

[2] R. S. PIERCE, Existence and uniqueness theorenis for extensions of zero-dimensional compact metric spaces, Transactions of the American Mathematical Society, vol. 148 (1970), pp. 1-21.

[3] M. REICHBACH, A note on O-dimensional compact sets, Bulletin of the Research Council of Israel, Section F (Mathematics and Physics), vol. 7F (1958), pp. 117-122.

UNIVERSITY OF HAWAII

HONOLULU, HAWAII 96822

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