On the continuous analogue of the Szpilrajn Theorem I

Mathematical Social Sciences 43 (2002) 115–134 www.elsevier.com / locate / econbase

On the continuous analogue of the Szpilrajn Theorem I Gerhard Herden*, Andreas Pallack Fachbereich 6 ( Mathematik /Informatik), Universitaet /GH Essen, Universitaetsstrasse 3, D-45117 Essen, Germany Received 1 October 2000; received in revised form 1 May 2001; accepted 1 June 2001

Abstract One of the best known theorems in order theory, mathematical logic, computer sciences and mathematical social sciences is the Szpilrajn Theorem which states that every partial order can be refined to a linear order. Since in mathematical social sciences one frequently is interested in continuous linear orders or preorders, in this paper the continuous analogue of the Szpilrajn Theorem will be discussed.  2002 Elsevier Science B.V. All rights reserved. Keywords: Continuous relation; Representation theorem; Separable system JEL classification: C00; D11

1. Introduction

1.1. The problems Let X be the consumption set of an economic agent and let R be an arbitrary binary relation on X. Then R is said to be order-like if for every point x [ X the corresponding indifference class I(x)[hy [ Xu(x, y) [ R ∧ ( y, x) [ Rj consists of at most one point. In addition, we set R S [h(x, y) [ Ru( y, x) [ ⁄ Rj. In order to make these concepts more transparent the reader may recall that an order is a reflexive, antisymmetric and transitive relation on X and that a preorder is a reflexive and transitive relation on X. This means that an order on X is an order-like preorder on X. This observation motivates our concept of an order-like relation on X. In addition, for any preorder ) on X the relation R S corresponds to a [h(x, y) [ )u¬(x | y)j. *Corresponding author. Tel.: 149-201-183-2516; fax: 149-201-183-2426. E-mail address: [email protected] (G. Herden). 0165-4896 / 02 / $ – see front matter  2002 Elsevier Science B.V. All rights reserved. PII: S0165-4896( 01 )00077-4

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In case that an order or preorder ) on X is known one may apply the classical Szpilrajn Theorem (cf. Szpilrajn (1930)) in order to refine ) to a linear order or preorder. Indeed, the Szpilrajn Theorem states that every order A on X has a linear refinement, i.e. there exists a linear order # on X such that A , # . In case that a preorder ) on X is given one may consider the corresponding set Xu| [h[x]ux [ Xj of indifference (equivalence) classes in order to get with the help of the Szpilrajn Theorem a linear order # on Xu| that refines ) u| . Then, by setting x & y⇔[x] # [y] for all points x, y [ X, the desired linear refinement of ) is defined. Since the concept of an extension of a preference relation was used by Jaffray (1975) and Yi (1991) in a different context, in contrast to the usual notation in the literature, we do not speak of an extension of ) by a linear order or preorder. Applying the Lemma of Zorn the Szpilrajn Theorem easily can be verified (cf., for instance, Erne´ (1982, Satz 10.13 and Korollar 10.14)). The original proof of Szpilrajn uses Zermelo’s Well-Ordering Theorem which is equivalent to the Lemma of Zorn. In case that X is endowed with some topology t one mainly is interested in continuous linear orders or preorders instead of only linear orders or preorders (the reader may recall that a linear order or preorder & on some topological space (X, t) is continuous if for every point x [ X both sets d(x)[hy [ Xuy & xj and i(x)[hz [ Xux & zj are closed subsets of X). This motivates the problem of generalizing the Szpilrajn Theorem to the continuous case. In order to be as general as possible we do not restrict our discussion of possible generalizations of the Szpilrajn Theorem to orders or preorders but also consider arbitrary binary relations or order-like binary relations on X. Clearly, one cannot expect that every binary relation or order-like binary relation on X can be refined to a continuous linear preorder or order. Refinable relations also must satisfy some kind of continuity condition. In order to approach the continuous analogue of the Szpilrajn Theorem, thus, in a first step the concept of continuity has to be generalized from linear preorders to arbitrary binary relations on X. Then four natural problems have to be discussed. Problem 1. Let R be an order-like binary relation on X. Determine necessary and sufficient conditions for the existence of some continuous linear order # on X such that R, #. # is said to be a continuous linear order refinement of R. In case that R is the equality relation ‘5’ on X Problem 1 reduces to the problem of characterizing orderable topological spaces. Therefore, Problem 1 is of particular interest. The reader may recall that a topological space (X, t) is said to be orderable if it can be endowed with a continuous linear order, and completely orderable if it can be endowed with a continuous linear order in such a way that the corresponding order topology coincides with t. Problem 2. Let R be a binary relation on X. Determine necessary and sufficient conditions for the existence of some continuous linear preorder & on X such that R , & and R S , , .

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In analogy to (1) & is said to be a continuous linear refinement of R. The reader may notice that the particular structure of Problem (1) and Problem (2), respectively, implicitly implies some kind of continuity condition to be satisfied by R (cf. Section 2.1). This means that continuity of R has not to be explicitly required in the assumptions of Problems (1) and (2), respectively. In addition, the reader may notice that in case of arbitrary binary relations on X the definition of a linear refinement is somewhat more complicated. Indeed, the additional inclusion R S , , has to be required. In case of order-like binary relations R on X our assumption on the indifference classes of R which generalizes the antisymmetry condition for orders guarantees that the inclusion R , # implies that also the inclusion R S , , holds. Problem 3. Determine necessary and sufficient conditions for a topology t on X to have the Szpilrajn property, i.e. every continuous order-like binary relation on X has a continuous linear order refinement. Problem 4. Determine necessary and sufficient conditions for a topology t on X to have the weak Szpilrajn or refinement property, i.e. every continuous binary relation on X has a continuous linear refinement. This paper is merely dedicated to the discussion of Problems (1) and (2). Indeed, the discussion of these problems continues some work of the first author (cf. Herden, 1989a,b, 1995). Problems (3) and (4) will be discussed in Herden and Pallack (2001). The Szpilrajn Theorem or its generalizations by Dushnik and Miller (1941) and, more recently, by Donaldson and Weymark (1998), Bossert (1999), Duggan (1999) and others is one of the most quoted theorems in order theory, mathematical logic, computer sciences, mathematical social sciences and other fields of pure and applied mathematics. Its relevance in order theory is well known. In a recent paper by Fishburn (1998) possible generalizations of the Szpilrajn Theorem to additive partial orders are discussed. Felgner and Truss (1999) consider the Szpilrajn property or order-extension property, which means that every partial ordering can be extended or refined to a linear ordering, in order to discuss the question if the Szpilrajn property implies the prime ideal theorem which states that every boolean algebra has a prime ideal. In Petri’s concurrency theory one usually works with a discrete version of Dushnik’s and Miller’s generalization of the Szpilrajn Theorem. In their paper on modelling concurrency Janicki and Koutny (1997, Theorem 2.9) characterize extension complete order structures and, thus, generalize Dushnik’s and Miller’s version of the Szpilrajn Theorem. Stehr (1996) applies the Szpilrajn Theorem in his characterization of global orientability. In addition, the Szpilrajn Theorem and generalizations of it are of importance in social choice theory (cf., for instance, the paper of Nehring and Puppe (1998) on a unifying structure of abstract choice theory or Sholomov (2000) who uses an analogue of the Szpilrajn Theorem in order to characterize ordinal relations). Diaye (1999, Lemma 1 and Proposition 1 of Section 3) applies the Szpilrajn Theorem in order to prove a nice result on the representation of acyclic relations. Weymark (2000) applies Dushnik’s and Miller’s generalization of the Szpilrajn Theorem in order to prove a generalization of Moulin’s Pareto extension theorem (cf. Moulin, 1985). Finally, we still want to

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underline the general importance of Szpilrajn-like theorems by quoting the paper of Suck (1994) where the Szpilrajn Theorem is used in order to prove a theorem on the applicability of conjoint structures. The preferences of an economic agent do not necessarily satisfy linearity. Nevertheless, also in this situation one is interested in minimal or maximal sets (cf., for instance, Pattanaik and Banarjee, 1996). In some situations preferences even cannot be described by a preorder. One may think, for instance, of interval orders that are frequently used in choice theory. The recent paper of Oloriz et al. (1998) discusses the problem of representability of interval orders in full generality. In addition, the reader may consult ´ the survey articles by Alcantud and Gutierrez (1997a,b). Therefore, we also consider arbitrary binary relations on X. In addition, in mathematical social sciences, in particular in mathematical utility theory, the consumption set of an economic agent usually is endowed with some topological structure. This means that continuous utility representation theorems that consider this situation are of importance. The first more general continuous utility representation theorem for a non-linear preference relation was proved by Peleg (1970). Peleg only assumes that a strict partial order on the consumption set is given. General continuous utility representation theorems of Peleg type were also proved by Levin (1983a,b), Mehta (1986, 1988) and Herden (1989a,b). In Herden (1995) a continuous utility representation theorem was proved that applies for arbitrary binary relations on X. On the other hand, a binary relation or order-like binary relation on some topological space is not necessarily representable. Therefore, some continuous analogue of the Szpilrajn Theorem is of interest. Indeed, let R be a binary or order-like binary relation on a compact space (X, t). Let us assume that R has a continuous linear refinement &. Then the compactness of (X, t) implies that & is a complete linear preorder or order. This means, in particular, that & has first (minimal) and last (maximal) elements. Of course, any minimal element of & is also a minimal element of R while any maximal element of & also is a maximal element of R, which underlines the importance of the problems (1)–(4) that have been presented in Section 1.1. Problems of Szpilrajn-type include the general continuous utility representation problem. In order to discuss this problem we consider the views of Chipman (1960). Chipman argues (cf., for instance, pp. 210–211 and pp. 221–223) that there are strong reasons for not choosing R as the codomain of a utility function on a preordered set because utility is inherently of lexicographic nature. Indeed, in a sense, the non-existence of a real-valued utility function reflects in some degree the properties of the given binary relation, but is also partly a consequence of a probably injudicious choice of the codomain. In the opinion of the authors the choice of an appropriate codomain of a (continuous) utility function depends on the particular situation to be considered. In a very general sense, a binary relation R on a topological space (X, t) has a continuous utility representation if there exists some topological space (Y, # , t) that is endowed with a continuous linear order # and, in addition, some continuous order preserving function u on X the codomain of which is Y. One easily verifies that the problem of the existence of (Y, # , t) and u is equivalent to Problem (2). Thus, Szpilrajn-type theorems are also within the main stream of efforts that have been made in order to completely clarify the differences between preference and utility.

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2. Continuous binary relations

2.1. The definition Let (X, R, t) be some R-space, i.e. a topological space that is, in addition, endowed with some arbitrary binary relation R on X. Then a subset S of X is said to be decreasing if y [ S and xRy imply that x [ S. By duality one defines the concept of an increasing subset S of X. In addition, a real-valued function f on X is said to be increasing if xRy implies that f(x) # f( y). With the help of this notation the reader may recall from Nachbin (1965) and Herden (1995) that (X, R, t) is said to be R-normal if for any pair of disjoint closed decreasing, respectively increasing, subsets C, D of X there exist disjoint open decreasing, respectively increasing, subsets U, V of X such that C , U and D ,V. Let t nat denote the natural topology on R. Then a straightforward generalization of the Nachbin Separation Theorem states that in order for (X, R, t) to be R-normal it is necessary and sufficient that for every pair C, D of disjoint closed decreasing, respectively increasing, subsets of X there exists some continuous increasing function f : (X, R, t) → ([0, 1], # , t nat ) such that f(C) 5 h0j and f(D) 5 h1j (cf. Herden, 1995). The original Nachbin Separation Theorem is restricted to preorders. Our concept of a continuous binary relation on X is based upon the following lemmas. Lemma 2.1. In order for R to have a continuous linear refinement it is necessary that for every pair (x, y) [ R S there exists some continuous increasing real-valued function f on X such that f(x) , f( y). Proof. Let & be a continuous linear refinement of R. Lemma 5.2.1 in Bridges and Mehta (1995) (cf. also Mehta, 1986) implies that (X, & , t) is a &-normal space. It follows from the continuity of & that d(x) and i( y) are for every pair (x, y) [ , closed disjoint decreasing, respectively increasing, subsets of X. The Nachbin Separation Theorem, thus, guarantees for every pair (x, y) [ , the existence of some continuous increasing real-valued function f on X such that f(x) , f( y). Since & is a refinement of R we may conclude that for every pair (x, y) [ R S there exists some continuous increasing real-valued function f on X such that f(x) , f( y) and the desired conclusion follows. h Lemma 2.2. In order for a linear preorder & on X to be continuous it is necessary and sufficient that for every pair (x, y) [ , there exists some continuous increasing real-valued function f on X such that f(x) , f( y). Proof. Because of Lemma 2.1 it suffices to verify the sufficiency part of the lemma. Let, therefore, the assumption of the lemma be satisfied and let some point x [ X be arbitrarily chosen. We must show that both sets d(x) and i(x) are closed subsets of X. Clearly, we only have to show that d(x) is a closed subset of X. Then it follows by duality that also i(x) is a closed subset of X. In order to prove that d(x) is a closed subset of X we may assume without loss of generality that K(x)[hy [ Xux , yj is neither empty nor contains a minimal element z. Indeed, if K(x) is empty, then d(x) 5 X is a

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closed subset of X. If K(x) contains some minimal element z, then we may consider some arbitrary continuous increasing real-valued function f on X such that f(x) , f(z) in order to conclude that d(x) 5 f 21 (] 2 `, f(z)]) is a closed subset of X. Therefore,we may proceed by choosing for every point y [ K(x) some point u [ K(x) such that u , y and some continuous increasing real-valued function fuy on X such that fuy (u) , fuy ( y). Then 21 the properties of the functions fuy imply that d(x) 5 > y [K(x) f uy (] 2 `, fuy (u)]) is a closed subset of X, which still was to be shown. h Lemmas 2.1 and 2.2 motivate the following definition. Definition 2.3. R is said to be continuous if for every pair (x, y) [ R S there exists some continuous increasing real-valued function f on X such that f(x) , f( y). Every binary relation R on X is contained in the trivial continuous linear preorder T [X 3 X on X. Hence, for every binary relation R on X the set Lc (R) of all continuous linear preorders & on X such that R , & is not empty. In addition, every continuous real-valued function f on X induces the linear continuous preorder & f on X that is defined by setting x & f y⇔ f(x) # f( y) for all points x, y [ X. Definition 2.3, thus, implies with the help of Lemma 2.1 and Lemma 2.2 the following characterization of a continuous binary relation on X that underlines the intimate relation between arbitrary continuous binary relations and continuous linear preorders on X. Proposition 2.4. In order for a binary relation R on X to be continuous it is necessary and sufficient that R S , < h , u & [ Lc (R)j.

2.2. R–t-topologies Let R be some binary relation on an arbitrarily but fixed chosen topological space (X, t). For every subset S of X we denote by S¯ its topological closure. Then we consider the set FR (X) of all continuous increasing real-valued functions f on X in order to define the R–t-topology t R on X as the weak topology t (X, FR (X)), i.e. t (X, FR (X)) is the coarsest topology on X for which every function f [ FR (X) is continuous. Let & be an arbitrary preorder on X. Then the reader may recall that a sub-basis of the order topology t & on X is given by 5, X and the families L[hL(x)[hy [ Xu y a xjj x [X and K[hK(x)[hz [ Xux a zjj x [X . Now the following lemma justifies the concept of an R–t-topology on X. Lemma 2.5. Let & be a linear preorder on (X, t). Then the following assertions are equivalent: (i) & is continuous. (ii) t & , t. (iii) t & 5 t & . Proof. The equivalence of the assertions (i) and (ii) is well known. Since every weak

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topology on X is coarser than t it, thus, remains to verify the validity of the implication ‘(ii) ⇒ (iii)’. The inclusion t & , t & is an immediate consequence of the definitions of t & and t & , respectively. Hence, only the inclusion t & , t & has to be shown. But the proof of this inclusion is implicitly in the proof of Lemma 2.2. h On the other hand, in case that & is an arbitrary preorder or order on X, then the inclusion t & , t does not necessarily imply that & is continuous as the following example shows. Example 2.6. Let X[h1, 2, 3j, t[h5, h1j, h3j, h1, 3j, Xj and #[h(1,1), (2,2), (3,3), (1,3)j. Then it follows that t # 5 t and t # 5 h5, Xj, which, in particular, means that t # , t and that # is not continuous. In addition, let A be an arbitrary preorder or order on X such that for every point x [ X both sets d(x) and i( y) are closed. Then the following example shows that A is not necessarily continuous. Example 2.7. Let X be an arbitrarily uncountable set. We consider the topology t[ h5, Xj < hX\EuE is a finite subset of Xj, choose two arbitrary points u, v [ X and define an order & on X by setting A[h(x, x)ux [ X) < h(u, v)j. Since the meet of every pair of non-empty open subsets of X is non-empty any continuous real-valued function f on X is constant. Hence, A is not continuous. On the other hand, (X, t) is a T 1 -space and we may conclude with the help of the definition of A that for every point x [ X both sets d(x) and i( y) are closed. In contrast to the Examples 2.6 and 2.7 the following proposition holds. The reader may notice that its proof is based upon arguments that already have been applied by Urysohn (1925), Nachbin (1965) and Peleg (1970). Proposition 2.8. Let ) be a preorder on (X, t) that satisfies the following conditions: C1: t ) , t. C2: for every point x [ X both sets d(x) and i(x) are closed. Then ) is continuous. Proof. Let x, y be arbitrary points of X such that x a y. Then we consider the linearly ordered set (Q I , # ) of all rationals in the real interval [0, 1] and set Bx, y [hL(z)uz [ [x, y]j. We may assume without loss of generality that there exists no pair of points u, v [ X such that x)u a v)y and ]u, v[ 5 ht [ Xuu a t a vj 5 5. Indeed, otherwise the conditions C1 and C2 imply that both sets d(u) and i(v) are open and closed and we may define, for instance, a continuous increasing real-valued function f on X such that f(x),f( y) by setting for every point z [ X

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f(z)[

H

0, 1,

if z [ d(x) otherwise

In case that there exists no pair of points u, v [ X such that x)u a v)y and ]u, v[ 5 5 a routine induction argument allows us to construct a subchain (C, )) of ([x, y],)) that is order-isomorphic to (Q I , # ). With the help of the conditions C1 and C2 we, thus, may conclude that there exists some function g: (Q I , # ) → (Bx, y , , ) such that ]] g( p) ,g( p) , g(q) for all pairs of points p , q [ Q I . The desired real-valued function f on X, thus, is defined by setting for every point z [ X f(z)[

H

1,

if z [ ⁄ < g( p)

infh p [ Q I uz [ g( p)j,

otherwise

p[Q I

Since the sets g( p) are decreasing for every rational p [ Q I we may conclude that f is increasing. Furthermore, the definition of f implies that 0 5 f(x) , 1 5 f( y). Finally, the continuity of f follows in the same way as the continuity of the analogously defined functions in Peleg’s Theorem (cf. Peleg, 1970). h The converse of Proposition 2.8 does not hold. Indeed, one easily verifies by the following example that there may exist continuous orders or preorders ) on X that do not satisfy any of the conditions C1 or C2 of Proposition 2.8. Example 2.9. Let (X, t) be defined as in Example 2.7. We consider the topological sum Y[X1 % X2 (both spaces X1 and X2 are homeomorphic to X) and choose arbitrary infinite sets S1 in X1 and S2 in X2 whose complements in X1 , respectively X2 , are also infinite. Then the desired order A on Y is the union of the set of all pairs (x, x) (x [ X) with the set of all pairs (u, v) (u [ S1 and v [ S2 ). The definitions of Y and A imply that A is continuous. Conversely, it follows that for every point u [ S1 and every point v [ S2 none of the sets d(v) or i(u) is closed and none of the sets K(u) or L(v) is open. Remark 2.10. Both conditions C1 and C2 are equivalent in the linear case. Condition C1 postulates compatibility of the order topology with t. Therefore, it is obvious to identify condition C1 with continuity of an arbitrary relation on X. On the other hand, because of the afore-discussed results and the results in Section 2.1 the authors think that the R-t-topology is more appropriate in order to study the existence of continuous utility representations than the order topology on X (cf., for instance, Section 2.3). Condition C2 of Proposition 2.8 is closely related to the frequently taken assumption ) to be closed (cf., for instance, Nachbin (1965) or Levin (1983a,b)), which means that ) is a closed subset of X 3 X endowed with the product topology. Indeed, it follows from the proof of Proposition 1 of Nachbin (1965) that a closed preorder on X always satisfies condition C2 of Proposition 2.8. The converse does not hold. For example, the equality relation ‘5’ on some topological space (X, t) is closed iff (X, t) is a Hausdorff-space, and satisfies condition C2 of Proposition 2.8 if (X, t) is a T 1 -space. On

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the other hand, a linear preorder on X is closed if it satisfies condition C2 of Proposition 2.8. Let R be a continuous binary relation on X. Then a closed continuous preorder ) on X such that R , ) and R S , a is said to be a continuous refinement of R by a closed preorder. With the help of the above remark and notation, despite Example 2.9, the following propositions hold. Proposition 2.11. Every continuous binary relation R on X has a continuous refinement by a closed pre-order. Proof. Let R be some continuous binary relation on X. Then we denote by F(R) the set of all continuous increasing real-valued functions f on X, and set )[h(x, y) [ X 3 Xu; f [ F(R)( f(x) # f( y))j. Clearly, ) is a continuous preorder on X such that R , ) and R S , a . It, thus, suffices to verify that ) is closed. Let, therefore, (u, v) be some pair of X 3X that is not contained in ). Then there exists some function f [ F(R) such that f(u) . f(v). The continuity of f implies the existence of neighborhoods U of u and V of v such that f(s) . f(t) for every pair (s, k) [ U 3V. Hence, (U 3V ) > ) 5 5 and the desired conclusion follows. h Proposition 2.12. Let (X, t) be a compact (Hausdorff-)space or a locally compact second countable (Hausdorff-)space that is endowed with some closed preorder ). Then ) is continuous. Proof. In case that (X, t) is a compact (Hausdorff-)space the desired conclusion follows from the proofs of Propositions 1, 4 and 5 and Theorem 4 of Nachbin (1965). The reader may notice that the proofs of these results do not depend on the assumption ) to be an order. In case that (X, t) is a locally compact second countable (Hausdorff-)space one may consult, for instance, the first (short) part of the proof of Lemma 8.3.4 in Bridges and Mehta (1995). This part of the proof is essentially based upon the compact version of Nachbin’s Lifting Theorem (cf. Nachbin (1965, Theorem 6)). Also the proof of this theorem does not depend on the assumption ) to be an order. A weaker version of this theorem was proved by Levin (1983a). h We still want to clarify in this subsection if for every continuous order or preorder ) on X such that a ±5 the )-topology t ) is necessarily )-normal. Example 2.13. Let # be an arbitrary continuous order on X such that , ±5. Then t # is not necessarily #-normal. Indeed, let t N be the well known Niemytzki-topology on R3R $0 (cf., for instance, Steen and Seebach, 1978). Then we choose two arbitrary points a, b [ ⁄ R 3 R $0 and consider the topology t on X[R 3 R $0 < ha, bj that is induced by t N and the singletons haj and hbj. Obviously, # [h(x, x)ux [ Xj < h(a, b)j is a continuous order on X such that , ±5. Since t is a completely regular Hausdorfftopology on X it follows that t and t # coincide. Hence, the definition of # implies that

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t 5 t # is #-normal iff t 5 t # is a normal topology on X. In addition, it follows from the definition of # that t 5 t # is a normal topology on X iff t N is a normal topology on R3R $0 . But it is well known that t N is not normal. Indeed, Definition 2.3 implies that for any continuous binary relation R on X the R–t-topology t R on X is R-seminormal, i.e. for every pair (x, y) [ R S for which there exists disjoint closed decreasing, respectively increasing, subsets C, D of X such that x [ C and y [ D there exists some continuous increasing real-valued function f on X such that f(x) , f( y). With the help of Lemma 2.5 and Lemma 5.2.1 in Bridges and Mehta (1995) we may conclude that for a linear preorder & on X the concepts of a &-normal topology and of a &-seminormal topology on X coincide.

2.3. On the generalizability of the Eilenberg–Debreu utility representation theorems The Eilenberg utility representation theorem states that every continuous linear preorder & on a separable connected topological space has a continuous utility representation (Eilenberg, 1941), i.e. there exists a continuous increasing real-valued function u on X such that u(x) , u( y) for every pair of points x, y [ X such that x , y, while the Debreu utility representation theorem states that every continuous linear preorder on a second countable topological space has a continuous utility representation (Debreu, 1954, 1964). The Eilenberg utility representation theorem cannot directly be generalized to arbitrary continuous orders or preorders as the following example shows. Example 2.14. We construct a space (X, A, t) with continuous order A which is separable and connected but which has no continuous utility representation. Therefore, we consider some uncountable set V 5 h0, 1, 2, . . . , v, v 1 1, v 1 2, . . . j of ordinals and the open interval ]0, 1[ of the reals. Then we set X[V < ]0, 1[. A sub-basis for a topology t on X is given by the family of all sets hg j < ]0,r[ and ]s,1[(g [ V, 0 , r,s , 1). The desired order A on X is defined by A[h(t, h )ut, h [ V, t # h j < h(r, s)ur, s [ ]0, 1[, r # sj. Trivially Q>]0,1[ is a countable dense subset of (X, t). Hence, (X, t) is separable. Furthermore, the connectedness of ]0,1[ immediately implies that also (X, t) is connected. In addition, the continuity of the natural linear ordering on ]0,1[ with respect to the natural topology on ]0,1[ implies with the help of the definitions of t and A that A is continuous. On the other hand, (X, A, t) has no (continuous) utility representation since (R, #) cannot contain a copy of (V, # ), which still was to be shown. In contrast to the Eilenberg utility representation theorem the Debreu utility representation theorem can directly be generalized to arbitrary continuous binary relations R on X. Theorem 2.15. Let (X, R, t) be a second countable topological space that is endowed with some continuous binary relation R. Then R has a continuous utility representation.

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Proof. We consider for every pair (x, y) [ R S some continuous increasing real-valued function fxy on X such that fxy (x) , fxy ( y) and set Ufxy (x)[ f 21 xy (] 2 `, fxy (x)[) and 21 Vfxy ( y)[ f xy (] fxy ( y), `[). Then the family C[hUfxy (x) 3Vfxy ( y)j (x, y)[R S is an open covering of R S . Since (X, t) is second countable there exists a countable subfamily C9 of C that also covers R S . This means, in particular, that there exists a countable family h fn j n[N\h0j of continuous increasing real-valued functions fn on X such that for every pair (x, y) [ R S there exists some n [ N\h0j such that fn (x) , fn ( y). Hence, u[o `n 51 (1 / 2 n) fn is a continuous utility representation of R. h Proposition 2.12 and Theorem 2.15 immediately imply the following interesting result that is essential in the proof of Bridges and Mehta’s version of Levin’s Theorem on jointly continuous utility functions (cf. Bridges and Mehta, 1995, Lemma 8.3.4; Levin, 1983a,b). Proposition 2.16. Let (X, t) be a second countable locally compact (Hausdorff-)space that is endowed with a closed preorder ). Then ) has a continuous utility representation.

2.4. A general continuous utility representation theorem Let X be some fixed given set. The reader may recall that a linear preorder ) on X is said to be: Debreu-separable if there exists a countable subset Z of X such that for every pair of points x, y [ X such that x , y there exists some point z [ Z such that x & z & y; Jaffray-separable if there exists a countable subset Z of X such that for every pair of points x, y [ X such that x , y there exists a pair of points z, z9 [ Z such that x & z , z9 & y. In addition, the reader may recall that a closed interval [x, y] of (X, & ) is said to be a jump of & if the corresponding open interval ]x, y[ of (X, & ) is empty. Let t be a fixed topology on X. Then the following theorem is well known (cf., for instance, Theorem 3.2.9 in Bridges and Mehta (1995)). Theorem 2.17. Let & be a linear preorder on X. Then the following assertions are equivalent: (i) & has a continuous utility representation. (ii) & is continuous and Debreu-separable. (iii) & is continuous and Jaffray-separable. (iv) & is continuous, t & is separable and & has only countably many jumps. The reader may notice that Theorem 2.17 includes the non-continuous case since any linear preorder & is continuous with respect to the discrete topology t dis on X, i.e. t dis is the power set P(X) of X. We want to generalize Theorem 2.17 to arbitrary binary relations R on X. Let, therefore, R be a binary relation on X. Then we set R(X)[hx [ Xu'y [ X((x, y) [ R ∨

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( y, x) [ R)j. As in the case of preorders we denote for each pair of points x, y [ X such that xRy by [x, y] the set (closed interval of (X, R)) of all points z [ X such that xRzRy. By ]x, y[ we denote, analogously, the set (open interval of (X, R)) of all points z [ X such that xR S zR S y. In addition, we denote for any subset Z of X by Z(R) the set of all closed intervals [x, y] of (X, R) such that ]x, y[ > Z 5 5. Finally, a binary relation P on X is said to be a countable refinement of R if R , P, R S , PS and P(X)\R(X) is countable. Now R is said to be Debreu-separable if there exists a countable refinement P of R and some countable subset Z of X such that for every pair of points x, y [ X such that xPS y there exists some point z [ Z such that xPzPy; Jaffray-separable if there exists a countable refinement P of R and some countable subset Z of X such that for every pair of points x, y [ X such that xPS y there exists some pair of points z, z9 [ Z such that xPzPS z9Py; almost separable if there exists a countable refinement P of R and some (countable) subset Z of X such that Z(P) is countable. In case that P is a continuous binary relation on X the underlying binary relation R on X is said to be continuously Debreu-separable, respectively continuously Jaffrayseparable, respectively continuously almost separable. The reader may verify that the general concepts of Debreu-separability and Jaffrayseparability, respectively, are in the linear case equivalent to the corresponding concepts of Debreu-separability and Jaffray-separability, respectively. In addition, the concept of almost separability is in the linear case equivalent to the conditions t & to be separable and & to only have countably many jumps. Now the following generalization of Theorem 2.17 holds. Theorem 2.18. Let R be an arbitrary binary relation on X. Then the following assertions are equivalent: (i) R has a continuous utility representation. (ii) R is continuously Debreu-separable. (iii) R is continuously Jaffray-separable. (iv) R is continuously almost separable. Proof. For the sake of brevity we only prove the equivalence of the assertions (i) and (iv). The equivalences of the assertions (i) and (ii) and (i) and (iii) are implicitly in the proof of the equivalence of assertions (i) and (iv). (i) ⇒ (ii): Lemma 2.1 implies with the help of Definition 2.3 that R is continuous. Let now f : (X, R, t) → (R, # , t nat ) be a continuous utility representation of R. We consider the linear preorder & f on X that is induced by f, i.e. x & y⇔ f(x) # f( y). Assertion (iv) of Theorem 2.17 implies the existence of some countable subset Z of X such that for every pair of points x, y [ X such that x , y and ]x, y[ ± 5 there exists some point z [ Z such that x , z , y. Now we set P0 [R and construct P inductively. Let, therefore, n $ 0 be some natural number. Then we choose arbitrary points x, y [ X such that xPn S y and consider the corresponding closed interval [x, y] of (X, & f ). In case that ]x, y[ ± 5 we may choose some fixed point z [ Z such that x , z , y in order to consider both pairs (x, z) and (z, y). Now Pn11 consists of Pn and all pairs (x, z) and (z, y) respectively that

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have been constructed in the afore-described way. Clearly, f also is a continuous utility representation of Pn 11 . Finally we set P[ < n[N Pn . The construction of P implies that f is a continuous utility representation of P and that P is a countable refinement of R. It, thus, remains to show that Z(P) is countable. Let, therefore, x, y [ X be arbitrarily chosen points such that xPS y and ]x, y[ 5 5. Because of the construction of P it follows that the corresponding closed interval [x, y] of (X, & f ) is a jump of & f . Assertion (iv) of Theorem 2.17 says that & f only has countably many jumps. Hence, Z(P) is countable. (iv) ⇒ (i): let P be a continuous binary relation on X for which there exists some (countable) subset Z of X such that Z(P) is countable. In order to prove that assertion (i) follows from assertion (iv) it suffices to verify that P has a continuous utility representation. Since Z(P) is countable there exists a countable family h(z n , z 9n )j n [N\h0j of pairs (z n , z 9n ) [ PS such that for every pair (x, y) [ PS there exists some n [ N\h0j such that xPz n PS z n9 Py. The reader may notice that this last conclusion corresponds to the assumption of assertion (iii). The continuity of P allows us to conclude that for every n [ N\h0j there exists a continuous increasing real-valued function fn on X such that ` fn (z n ) , fn (z 9n ). Hence, u[o n51 1 / 2 n fn is a continuous utility representation of P, which finishes the proof of the theorem. h

3. Discussion of Problems (1) and (2)

3.1. R-separable systems on X Let (X, R, t) be some fixed given R-space. In order to show that many of the apparently disparate approaches to mathematical utility theory incorporated in the theorems of Eilenberg (1941), Debreu (1954, 1964), Peleg (1970), Arrow and Hahn (1971) and others are in fact special cases of a general theory of order preserving functions on topological preordered spaces in Herden (1989a) the concept of an R-separable system on X was introduced. Definition 3.1. A family E of open decreasing subsets of X is said to be an R-separable system in the sense of Peleg on X (briefly: R-separable system on X) if it satisfies the following conditions: PS1: there exist sets E1 , E2 [ E such that E¯ 1 , E2 . PS2: for all sets E1 , E2 [ E such that E¯ 1 , E2 there exists some set E3 [ E, such that E¯ 1 , E3 , E¯ 3 , E2 . E is said to be linear if E , E9 or E9 , E for every pair of sets E, E9 [ E. The E-topology t E of some linear R-separable system E on X is induced by the sets E and X\E¯ where E runs through E. In case that R is the equality relation ‘5’ on X, i.e. the discrete order on X, then we speak of a separable system on X in the sense of Urysohn (briefly: separable system on X).

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The reader may verify that in case that t is a normal topology on X it follows that t is a separable system on X. The converse assertion does not hold in general (cf. Herden and Mehta, 2001, Proposition 1.1.6). In mathematical utility theory R-separable systems on X were constructed for the first time by Peleg (1970) in order to prove his utility representation theorem. Although Peleg did not see the intimate relation between his construction and the earlier constructions due to Urysohn and Nachbin this is the reason why we also speak of an R-separable system on X in the sense of Peleg instead of merely an R-separable system on X. Burgess and Fitzpatrick (1977) studied decreasing scales in X. A family S[hFr j r [D of open decreasing subsets of X is said to be a decreasing scale in X if the following conditions are satisfied: DS1: D is a dense subset of [0,1] such that 1 [ D and F1 5 X. DS2: for every pair of real numbers r 1 , r 2 [ D the inclusion F¯ r 1 , Fr 2 holds. One easily verifies that decreasing scales in X are particular cases of linear Rseparable systems on X. In the Introduction it has been emphasized that Problem (2) may be considered as the most general version of the continuous utility representation problem. In case that the codomain of a utility representation is the real line in Herden (1995) the following theorem was proved (cf. also Bridges and Mehta, 1995, Theorem 5.2.12), which in combination with Theorem 2.18 completely solves the continuous utility representation problem in case that we require the codomain of a utility function to be the real line. Theorem 3.2. The following assertions are equivalent: (i) R has a continuous utility representation. (ii) There exists a countable linear R-separable system E on X such that for every pair (x, y) [ R S there exists some pair of sets E, E9 [ E such that E¯ , E9, x [ E and y [ X\E9. (iii) There exists a countable family hEn j n[N of R-separable systems En on X such that for every pair (x, y) [ R S there exists some n [ N such that x [ E and y [ X\E for every set E [ En . Theorem 3.2 motivates the aim of generalizing assertions (ii) and (iii) of the theorem in order to prove a general continuous utility representation theorem that initially solves Problem 2 but also is the basis for solving Problem 1.

3.2. Solution of Problem 2 Let (X, R, t) be a fixed chosen R-space and let E be some R-separable system on X. Then we denote by ES the set of all pairs (x, y) [ X 3 X for which there exists some pair of sets E, E9 [ E such that E¯ , E9, x [ E and y [ X\E9. The solution of Problem 2 is based upon the following definition.

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Definition 3.3. A family hEi j i [I of R-separable systems Ei on X is said to be compatible if for every index i [ I and every pair (x, y) [ Ei S the set of all indexes j [ I such that ( y, x) [ Ej S is at most countable. Theorem 3.4. The following assertions are equivalent: (i) R has a continuous linear refinement. (ii) There exists a linear R-separable system E on X such that for every pair (x, y) [ R S there exists some pair of sets E, E9 [ E such that E¯ , E9, x [ E and y [ X\E9. (iii) There exists a compatible family hEi j i [I of R-separable systems Ei on X such that for every pair (x, y) [ R S there exists some index i [ I 991 such that x [ E and y [ X\E for every set E [ Ei . Proof. (i) ⇒ (ii): let & be a continuous linear refinement of R. Then one immediately verifies that L[hL(x)j x [X 5 hhy [ Xuy , xjj x [X is a linear R-separable system on X that satisfies the assumptions of assertion (ii). (ii) ⇒ (iii): let E be a linear R-separable system on X that satisfies the assumptions of assertion (ii). Then we may conclude with the help of condition PS2 and a routine induction argument that for every pair (x, y) [ R S there exists a countable linear R-separable system E(x, y) on X such that x [ E and y [ X\E for every set E [ E(x, y) . In addition, the linearity of E implies that the, in this way, obtained family hE(x, y) j (x, y)[R S of R-separable systems on X is compatible. (iii) ⇒ (i): the proof of this implication is based upon the idea to construct (by transfinite induction) a family F of continuous increasing real-valued functions f on X that satisfies the following conditions: Com1: for every pair (x, y) [ R S there exists some function f [ F such that f(x) , f( y). Com2: if f 9(u) , f 9(v) for some function f 9 [ F and some pair of points u, v [ X, then f(u) # f(v) for every function f [ F. For every pair of points s, k [ X we then set s & k⇔; f [ F ( f(s) # f(k)). Since F is a family of continuous increasing real-valued functions on X that satisfies the conditions Com1 and Com2 we may conclude that & is a continuous linear refinement of R. Hence, it suffices to construct F. The assumptions of assertion (iii) imply with the help of condition PS2 that we may assume without loss of generality that for every index i [ I the corresponding R-separable system Ei on X is countable and linear. Because of Theorem 3.2 there exists for every index i [ I some continuous increasing real-valued function fi on X that satisfies the following conditions: (*) for every pair (x, y) [ R S such that x [ E and y [ X\E for every set E [ Ei the inequality fi (x) , fi ( y) holds. (**) if fi (u) , fi (v) for some pair of points u, v [ X, then (u, v) [ Ei S .

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We, thus, choose the set H of all these functions fi , set K0 [5, F0 [5 and consider some arbitrary ordinal 0 , a , uHu. We may assume that H\ < b , a Kb ± 5. Otherwise, the induction process stops and we set F[ < b , a Fb . Let h [ H\ < b , a Kb be arbitrarily chosen. Then we set G0 [hhj and proceed by induction. Therefore, we assume that for some n [ N the set Gn , H\ < b , a Kb of functions already has been constructed. Because of the induction hypothesis we may assume that Gn is a countable set and that for every ordinal b , a, every function k [ H\Kb < Fb and every pair of points u, v [ X the following implication holds: ; f [ Fb (k(u) , k(v) ⇒ f(u) # f(v)). These implications may be summarized for the conclusion that for every function k [ (H\ < b , a Kb ) < ( < b , a Fb ) and every pair of points u, v [ X the following implication holds: (*a ); f [ < Fb (k(u) , k(v) ⇒ f(u) # f(v)). b ,a

For every function s [ Gn the continuous linear preorder & s that is induced by s is Jaffray-separable. This means that there exists some countable family h(z sn , z n9 s )j n [N of pairs (z sn , z 9s n ) [ , s such that for every pair (u, v) [ , s there exists some n [ N such that u & s z 9n , z 9n s & s v. The assumptions of assertion (iii) imply with the help of (*a ) that for n [ N every set G ns of functions g [ H such that g(z ns ) , g(z n9 s ) or g(z n9 s ) , g(z sn ) is a countable subset of H\ < b , a Kb . We, thus, set Gn 11 [Gn < ( < s [Gn < n[N G ns ). Gn11 is, obviously, a countable subset of H\ < b , a Kb . Finally, we set G[ < n[N Gn . Then also G is a countable subset of H\ < b , a Kb . Hence, there exists an enumeration hg j j j [N\h0j of G, and we may define a continuous increasing real-valued function fa on X by setting fa [o `j 51 1 / 2 j g j . Then Ka is the union of the sets < b , a Kb and G while Fa is the union of < b , a Fb with the singleton h fa j. The inductive construction of fa implies that for every function k [ H\Ka < Fa and every pair of points u, v [ X the following implication holds: (*a 1 1); f [ Fa (k(u) , k(v) ⇒ f(u) # f(v)), which completes the inductive argument. In the last step we still set F[ < a Fa . Because of the assumptions of assertion (iii) and the implications (*a ) it follows that F satisfies the conditions Com1 and Com2, which still was to be shown. h Remark 3.5. The reader may verify that in Example 2.14 the constructed topological space (X, t) is a normal first countable and separable Hausdorff-space. In addition, the reader may prove that in Example 2.14 the defined order A on X is closed and gives (X, A, t) the structure of a A-normal or normally ordered space. Finally, it immediately follows that there cannot exist any linear A-separable system E on X such that for every pair (x, y) [ a there exist sets E, E9 [ E such that E¯ , E9 [ E, such that E¯ , E9, x [ E and y [ X\E9, which because of assertion (ii) of Theorem 3.4 means that A does not have a continuous linear refinement. Since (X, t) is a separable and connected topological space this last conclusion also follows from the Eilenberg utility representation theorem. Indeed, the Eilenberg utility representation theorem implies that

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a binary relation on a separable and connected topological space has a continuous linear refinement if it has a continuous utility representation.

3.3. Solution of Problem 1 Let (X, R, t) be some fixed given R-space. We want to determine necessary and sufficient conditions for R to have a continuous linear order refinement. In this case R must be order-like, which means that we do not have to explicitly require R to be order-like. Let a linear R-separable system E on X said to be separated if the E-topology t E on X is a Hausdorff-topology. Then the solution of Problem 1 is based upon the following lemmas. Lemma 3.6. Let E be a separated linear R-separable system on X. Then uE¯ \Eu # 1 for every set E [ E. Proof. Let E be some separated linear R-separable system on X. We assume, in contrast, that there exists some set E [ E such that uE¯ \Eu $ 2. Then there exist points x, y [ E¯ \E such that x ± y. Since E is separated there exist sets E1 , E2 , E3 , E4 [ E such that x [ E1 > X\E¯ 2 , y [ E3 > X\E¯ 4 and (E1 > X\E¯ 2 ) > (E3 > X\E¯ 4 ) 5 5. We may assume without loss of generality that E1 , E3 . In case that E3 , E1 an analogous argument works. Now the disjointness of E1 > X\E¯ 2 and E3 > X\E¯ 4 implies with the help of the linearity of E that E¯ 2 , E¯ 4 . Hence, we may conclude that E1 , E¯ 4 , which means that E¯ 1 , E¯ 4 . Since x [ E1 \E it follows with the help of the linearity of E that E , E1 and, therefore, that E¯ , E¯ 1 . Summarizing these considerations the chain E , E¯ , E¯ 1 , E¯ 4 has been constructed. Hence, (E3 > X\E¯ 4 ) > E 5 5. This contradiction proves the lemma. h Lemma 3.7. Let E be a separated linear R-separable system on X. Then E¯ , E9 or E¯ 9 , E for every pair of different sets E, E9 [ E. Proof. Let E be a separated linear R-separable system on X. Then we choose sets E, E9 [ E such that E ± E9. We may assume without loss of generality that EvE9. Otherwise, E9vE and an analogous argument can be applied. In order to prove the lemma it, thus, still remains to show that E¯ , E9. Therefore, we assume, in contrast, that E¯ is not contained in E9. Then Lemma 3.6 implies the existence of some point x [ X such that E¯ 5 E < hxj and E¯ 9 5 E9 < hxj. Let some point y [ E9\E be arbitrarily chosen. Since E is separated there exist sets F, F9 and H, H9 [ E such that x [ F > X\F¯ 9, y [ H > X\H¯ 9 and (F > X\F¯ 9) > (H > X\H¯ 9) 5 5. Then the linearity of E implies with the help of the relations x [ F, x [ ⁄ E9, x [ E¯ and x [ ⁄ F¯ 9 and Lemma 3.6 that E¯ 9 , F and ¯ ¯ F 9 , E. Hence, E9\E 5 E9 > X\E , F > X\F 9, which, in particular, means that y [ F > X\F¯ 9, a contradiction. Therefore, the desired inclusion E¯ , E9 holds. h

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Lemma 3.8. Every linear R-separable system E on X induces a linear preorder & on X which satisfies the following properties: L1: R , & L2: the order topology t & is coarser than t E . Proof. Let E be a linear R-separable system on X and let points x, y [ X be arbitrarily chosen. Since Lemma 3.8 already was proved in Herden (1989b, 1995) we only repeat, for later use, the definition of &. Therefore, we set Ey [hE [ Euy [ Ej and define & by setting x & y⇔Ey 5 5 ∨ ;E [ Ey ;E9 [ Ey ((E¯ > X\E9 ± 5) ∨ (x [ E9)) One immediately verifies that & can be divided into the following two less complicated subrelations: x , y⇔'E [ E'E9 [ E((E¯ , E9) ∧ (x [ E) ∧ ( y [ X\E9)) and x | y⇔¬(x , y) and ¬( y , x). h Let a family hEi j i [I of R-separable systems on X said to be strongly compatible if for any pair of indexes j, k [ I at least one of the inclusions Ej S , Ek S or Ek S , Ej S holds. Then Problem (1) can be solved with help of the afore-proved lemmas in an analogous way as Problem (2). Theorem 3.9. The following assertions are equivalent. (i) R has a continuous linear order refinement. (ii) There exists a separated linear R-separable system E on X such that for every pair (x, y) [ R S there exists some pair of sets E, E9 [ E such that E¯ , E9, x [ E and y [ X\E9. (iii) There exists a strongly compatible family hEi j i [I of R-separable systems Ei on X such that for every pair of different points x, y [ X there exists some index i [ I such that either x [ E and y [ X\E for every set E [ Ei or y [ E and x [ X\E for every set E [ Ei . Proof. (i) ⇒ (ii): this implication follows in the same way as the corresponding implication in the proof of Theorem 3.4. (ii) ⇒ (i): let E be a separated linear R-separable system on X that satisfies the assumptions of assertion (ii). Then we consider the linear preorder & on X that has been constructed in the proof of Lemma 3.8. The definition of & implies with help of the assumptions of assertion (ii) and the inclusions t & , t E , t that it suffices to verify that & is a linear order on X. Therefore, we choose arbitrary points x, y [ X such that x ± y. Since E is separated there exists sets E1 , E2 , E3 , E4 [ E such that x [ E1 > X\E¯ 2 , y [ E3 > X\E¯ 4 and (E1 > X\E¯ 2 ) > (E3 > X\E¯ 4 ) 5 5. We may assume without loss of

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generality that E1 , E3 . If E3 , E1 , then an analogous argument works. Now we show that x , y. Then assertion (i) will be proved. The disjointness of (E1 > X\E¯ 2 ) and (E3 > X\E¯ 4 ) implies with help of the inclusion E1 , E3 and the linearity of E that E2 , E4 . Hence, E¯ 2 , E¯ 4 which means that X\E¯ 4 , X\E¯ 2 and we may conclude that E1 , E¯ 4 . Now we distinguish between the following two cases. Case 1. E1 5 E¯ 4 . In this case E1 is open and closed and it follows that E¯ 1 , E1 , x [ E1 and y [ X\E1 . The definition of , in the proof of Lemma 3.8, thus, implies that x , y, which settles the first case. Case 2. E1 vE¯ 4 . Now Lemma 3.7 implies that E¯ 1 , E4 . Since x [ E1 and y [ X\E4 the definition of ,, therefore, also implies in this case that x , y. (i) ⇒ (iii): let # be a continuous linear order refinement of R. Then we consider for every point x [ X for which L(x) 5 hy [ Xuy , xj ± 5 the (linear) R-separable system Lx [hL( y)j y ,x on X. Obviously, the family of the, in this way, defined (linear) R-separable systems on X is a strongly compatible family of R-separable systems on X that satisfies the assumptions of assertion (iii). (iii) ⇒ (i): let hEi j i [I be a strongly compatible family of R-separable systems Ei on X that satisfies the assumptions of assertion (iii). Then the arguments that are already implicitly in the proof of the corresponding implication of Theorem 3.4 imply with the help of the assumptions of assertion (iii) that by x , y⇔'i [ I;E [ Ei ((x [ E) ∧ ( y [ X\E)) and x | y⇔¬(x , y) ∧ ¬( y , x) a continuous linear order refinement of R is defined. h

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