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SEPARABILITY, THE COUNTABLE CHAIN CONDITION AND THE LINDELOF PROPERTY IN LINEARLY ORDERABLE SPACES D. J. LUTZER AND H. R. BENNETT 1. Introduction. It is known that, in metric spaces, separability, the Lindelof property and the countable chain condition1 (abbreviated CCC) are each equivalent to second countability. Furthermore, any space which is separable or hereditarily Lindelof satisfies the CCC (though not conversely). In this paper, we examine separability, the Lindelof property and the CCC in linearly orderable topological spaces (abbreviated LOTS), i.e., spaces whose topology is the open interval topology of some linear ordering of the underlying set. We shall prove that if X is a LOTS, then (A) X satisfies the CCC if and only if X is hereditarily Lindelof; (B) X is separable if and only if X is hereditarily separable. Result (A) was obtained for connected LOTS by T. Inagaki ([5] and [6]) and somewhat later by G. Kurepa [9]. The methods of Inagaki and Kurepa rely heavily on connectedness; our proofs avoid this assumption. (Also, see Remark 4.3, below.) Result (B) was announced without proof by G. Kurepa in [9]; L. Skula provided a proof in [14], but our proof is shorter. The following notation will be convenient. If (X, ^) is a linearly ordered set and if a£X, then ]— =o, a[= {x£X|x<a} and ]— co, a] = {x£X|x^a|. The sets ]a, °° [ and [a, co [ are defined analogously. 2. The countable chain condition in LOTS.2 The following lemma is easily proved by transfinite induction. (2.1) Lemma. Suppose X is a LOTS satisfying the CCC and suppose FCJ, Let zEY. Then there are countable subsets P and Q of ] — oo , z] r\ Y and [z, oo [f~\ Y respectively such that if yEY, then there are points pEP and qEQ such that p^y^q. (2.2) Theorem. A LOTS satisfying the CCC is hereditarily Lindelof. Proof. Suppose that X is a LOTS satisfying the CCC. To show that X is hereditarily Lindelof, it will suffice to show that if V is any collection of open intervals in X, then there is a countable subcollecPresented to the Society, January 25, 1969 under the title Covering properties of linearly ordered spaces and Souslin's problem; received by the editors April 18,1969. 1 A space X satisfies the countable chain condition if any disjoint collection of open sets is countable. 2 The authors are indebted to Mary Ellen Rudin for suggesting the approach used in this section. 664 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use SEPARABILITY,THE COUNTABLECHAIN CONDITION tion llCTj 665 which covers the set UT) (cf. [4, p. 141, exercise B]). Let F=Ul3. For each x£ F, let J(x) = {y£ Y\ the set of all points of F lying between x and y can be covered by a countable subcollection of V}. It is clear that each set I(x) is open in X and that for any two points x,yEY, either I(x) = I(y) or else I(x)(~\I(y) = 0. Because A satisfies the CCC, the collecion $= {/(x)|x£ Y} is countable, say 3 = {7(x)|x£C} where C is some countable subset of Y. Fix x£C. Using (2.1), find countable subsets P(x) and Q(x) of I(x)C\]— oo, x] and 7(x)P\[x, oo [ respectively such that if y£/(x), then there are points pEP(x) and qEQ(x) such that p^y^q. By definition of I(x), ior each pEP(x) and g£<2(x) there are countable subcollections V(x, p) and V(x, q) of V which cover the sets YC\ [p, x] and Yf\[x, q] respectively. Let 1l = U{u(x, r)\ rEP(x)\jQ(x) and x£C}. Then It is a countable subcollection of V which covers Y. 3. Separability in LOTS. Lemma (3.1) follows from (2.2) above; alternatively, it may be deduced from Lemma (3.2) below (which can be proved by modifying the proof that a LOTS is normal given in [3, p. 39]). (3.1) Lemma. Suppose that X is a LOTS satisfying the CCC and that Y is a discrete subspace of X (discrete in the relative topology). Then Y is countable. (3.2) Lemma. Any LOTS is hereditarily (3.3) Theorem. Proof. collectionwise normal.3 A separable LOTS is hereditarily separable. Suppose that A is a separable LOTS and that iCJ. Let 1(A) = {a£yl| {a} is relatively open in A }. By (3.1), 1(A) is countable. Let D he a countable dense subset of X. Let S>= []r,s[\r,sE For each interval D,r < s, /£SD, I(A)\J{a(J)\JE'£>}- and An]r,s[^0}. choose a point a(J)EA(~\J. Let D(A) = Clearly, D(A) is a countable subset of A. To prove that D(A) is dense in A, it will suffice to show that if U is open in A and if Af\U^0, a£/(,4), then D(A)C\U^0. then aED(A)C\U. If a$I(A), Choose a£.4rw. If then either (i) ]— oo, a[^0 and ]x, a[P\yl 7^0 whenever x<a, or else (ii) ]a, oo [^0 and ]a, y[C\A 7^0 whenever y>a. We consider only the first case. Since U is a neighborhood of a in X, there is a point z<a such that ]z, a]QU. Applying (i), we choose * Added in proof. A detailed proof of this result by Professor L. A. Steen will appear in these Proceedings. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 666 D. J. LUTZER AND H. R. BENNETT points b £]z, a[f~\A;cE]b, a[(~\A; anddEJc, in X and since ]z, c[^09^}c, sE]c, a[f\D. Letting J=]r, [December a[C\A. Since D is dense a[, there are points r£]z, c[C\D and s[, we obtain an element of £>. Then a(J)E]z, a[nD(A)QUr\D(A). 4. Examples. In this section, we will attempt to show how our results can be applied in general topology. (4.1) Example. Souslin spaces. Recall that a Souslin space is a nonseparable LOTS which satisfies the CCC. It was recently proved that the existence of a Souslin space is independent of the axioms of set theory ([lO] and [15]). It follows easily from a result of Mary Ellen Rudin in [13] that if there is a Souslin space, then there is a compact, connected Souslin space. Together with our Theorem (2.2), this shows that Problem 4.7 of [l] cannot be answered affirmatively. Similarly, Problem 4.6 of [l] cannot be answered affirmatively, since Ponomarev [12] and Bennett [2] have proved that if there is a Souslin space, then there is a Souslin space with a point-countable base. (4.2) Example. Let X be the space i?x{0, l} with the lexico- graphic order topology: see [8] for details. (X is called the double line.) It is clear that X is a separable LOTS. Hence, X is hereditarily separable and hereditarily Lindelof. Let £ be the topology on R obtained by taking the collection {[a, b[\a, bER and a<b} as a base. The space (R, £), called the Sorgenfrey line, is homeomorphic to a subspace (the "top line of points") of the double line. It follows from (2.2) and (3.3) that (R, £) is hereditarily separable, hereditarily Lindelof and perfectly normal (cf. [8, p. 134, Example K]). It should be pointed out that (R, £) is not itself a LOTS [ll]. (4.3) Remark. We observe that the Souslin space with a pointcountable base mentioned in (4.1) cannot be (locally) connected by [2, Theorem 15]. Also, the double line of (4.2) cannot be embedded in a (locally) connected LOTS satisfying show that the results of Inagaki and certain applications. (4.4) Remark on a paper the CCC. These two facts Kurepa are inadequate for of F. B. Jones. Professor F. Burton Jones has asked the authors to point out an error which appeared in one of his papers [7]. The five conditions listed on p. 627 are not equivalent, even in a connected LOTS, as our examples and the example given in [8, p. 164, Example K] show. In any LOTS, Jones' property (2) is equivalent to the CCC and to the hereditary Lindelof property. In a connected LOTS, Jones' properties (1), (4) and (5) are equivalent. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 1969] SEPARABILITY,THE COUNTABLECHAIN CONDITION 5. Generalizations. number. 667 In this section, m denotes an infinite cardinal (5.1) Definition. A space X is m-separable ii X contains a dense subset having cardinality ^ m; X satisfies the m-chain condition if any disjoint collection of open subsets of X has cardinality ^ m; X is m-Lindelbf if every open cover of X has a subcover with cardinality =w. Using the methods of §2 and §3, we can prove (5.2) Theorem. A LOTS satisfies the m-chain condition if and only if it is hereditarily m-Lindeldf. (5.3) Theorem. An m-separable LOTS is hereditarily m-separable. Bibliography 1. A. V. Arhangel'skii, Mappings and spaces, Uspehi Mat. Nauk 21 (1966), no. 4 (130), 133-184= Russian Math. Surveys 21 (1966)no. 4,115-162. 2. H. R. Benett, Quasi-developable spaces, Dissertation, Arizona State University, Tempe, Ariz., 1968. 3. G. Birkhoff, Lattice theory, Amer. Math. Soc. Colloq. Publ., vol. 25, Amer. Math. Soc, Providence, R. I., 1961. 4. R. 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Dokl. 8 (1967), 765-768. 13. M. E. Rudin, Countable paracompactness and Souslin's problem, Canad. J. Math. 7 (1955),543-547. 14. L. Skula, Hereditary m-separability of an ordered space, Casopis Pest. Mat. 90 (1965), 451-454 (Czech.). 15. S. Tennenbaum, Souslin's problem, Proc. Nat. Acad. Sci. U.S.A. 59 (1968), 60-63. University of Washington and Texas Technological College License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use