05.dvi @ Applied General TopologyUniversidad Polit�ecnica de ValenciaVolume 1, No. 1, 2000pp. 61 - 81 Useful topologies and separable systemsG. Herden, A. PallackAbstract. Let X be an arbitrary set. A topology t on X issaid to be useful if every continuous linear preorder on X is rep-resentable by a continuous real{valued order preserving function.Continuous linear preorders on X are induced by certain familiesof open subsets of X that are called (linear) separable systems onX. Therefore, in a �rst step useful topologies on X will be char-acterized by means of (linear) separable systems on X. Then, ina second step particular topologies on X are studied that do notallow the construction of (linear) separable systems on X thatcorrespond to non{representable continuous linear preorders. Inthis way generalizations of the Eilenberg{Debreu theorems whichstate that second countable or separable and connected topologieson X are useful and of the theorem of Est�evez and Herv�es whichstates that a metrizable topology on X is useful, if and only if itis second countable can be proved.2000 AMS Classi�cation: 54F05, 91B16, 06A05.Keywords: Completely regular topology, weak topology, normal topology,short topology, countably bounded topology, countably bounded linear pre-order. 1. IntroductionA topology t on an arbitrary set X is said to be useful, if every continuouslinear (total) preorder - on X has a continuous utility representation, i.e. canbe represented by a continuous real{valued order preserving function (utilityfunction) (see [15]). Continuity of - means that the order topology t- inducedby - is coarser than t. Su�cient conditions for a topology t on X to be usefulare, for instance, given by the classical Eilenberg{Debreu Theorems (EDT) and(DT) ([9, 10, 11]). Necessary and su�cient conditions for a topology t on Xto be useful have been presented by the theorem of Est�evez and Herv�es (EHT)in case that t is a metrizable topology on X ([13], see also [6, 7]) Using theconcept of a useful topology t on X these theorems can be restated as follows: 62 G. Herden, A. Pallack(EDT) Every connected and separable topology t on X is useful.(DT) Every second countable topology t on X is useful.(EHT) A metrizable topology t on X is useful, if and only if t is second count-able.The aim of this paper is the characterization of all useful topologies t onX. A theorem which solves this problem, in particular, would generalize theEilenberg-Debreu Theorems and the Theorem of Est�evez and Herv�es. Mean-while Banach-spaces or, more generally, convex spaces are frequently studiedin mathematical utility theory. In the in�nite dimensional case these spacesmay fail to be second countable or separable. This means that continuousrepresentation of linear orderings (preference orderings) in these spaces is notguaranteed by the classical Eilenberg-Debreu Theorems. Therefore, a charac-terization of useful topologies is of particular interest in mathematical utilitytheory (cf. also Remark 6.8).2. A first approachLet throughout this section X be a �xed given set and let t be some topologyon X. For every subset A of X we denote by A its topological closure. Themost fundamental result that is known on useful topologies is DT. DT easilyimplies EDT (cf. [14, 16]) and also generalizes the su�ciency part of the The-orem of Est�evez and Herv�es. On the other hand, it is well known that secondcountability, in general, is not necessary for t to be useful (cf., for instance, theNiemitzki plane that is extensively discussed in [32]). Hence, in order to at leastapproximate second countability we consider linearly ordered subtopologies tlof t. tl is linearly ordered if it is linearly ordered by set inclusion.It is easily to be seen that second countability of t implies second countabilityof all its linearly ordered subtopologies tl. Indeed, let tl be a linearly orderedsubtopology of the second countable topology t. Then we choose a countablebase B of t and consider the countable subsetBl := �O 2 tl j9B 2 B(B � O ^8O0 2 tl(O0 $ O =) B 6� O0)) [�O 2 tl j9B 2 B(O = [B 6�O02tlO0) [f?;Xgof tl in order to immediately verify that Bl is a base of tl. Let us now assumethat all linearly ordered subtopologies tl of t are second countable and let - bea continuous linear (total) preorder on X. Then we consider the familyL := fL(x)gx2X := ffy 2 X jy � xggx2Xof open decreasing subsets of X. The linearly ordered subtopology tl of t thatis induced by L is second countable which means that there exists a countablesubset LB of L [ f?;Xg that is a base of tl. The countability of LB impliesthat the corresponding chain (LB;�) only has countably many jumps. Thereader may recall that a jump of (LB;�) is a pair of sets E $ E0 2 LB suchthat there exists no set E00 2 LB such that E $ E00 $ E0. By interposingthe rationals into the jumps of (LB;�) we, thus, obtain some chain (LeB;�)that extends (LB;�) and may, without loss of generality, be assumed to be Useful topologies and separable systems 63order-isomorphic to the chain ([0;1]Q;�) of all rationals in the real interval[0;1]. Let g: (LeB;�) �! ([0;1]Q;�) be some order-isomorphism. Then oneveri�es that f: (X;-) �! ([0;1]R;�) de�ned for all points x 2 X by f(x) :=supfg(L(y)) jL(y) 2 LB;y - xg is a continuous utility representation of -.Clearly, t is not necessarily second countable if all its linearly ordered sub-topologies tl are second countable. In order to obtain a counterexample oneonly has to choose some topology on the natural numbers that is not secondcountable. Hence, the above result does not only provide an alternative proofof DT but also generalizes DT.In order to also generalize EDT we consider the �rst in�nite ordinal !. Thenwe consider the family TC of all linearly ordered subtopologies tl of t that areinduced by some linearly (totally) ordered set (O;�) of open subsets of X thatsatisfy the following conditions:(LO1): 8O0 2 O(O0 � TO0$O2O O) or, equivalently, 8O0 2 O 8O 2O(O0 $ O =) O0 � O), and(LO2): ���nO 2 Oj SO3O0$O O0 $ O ^SO3O0$O O0 \ X n O 6= ?o��� � !.Now it follows that in case that t is a separable and connected topology onX every linearly ordered subtopology tl 2 TC of t must be second countable.Indeed, let some topology tl 2 TC be arbitrarily chosen. Then the separabilityof t implies that no chain (O;�) or (O;�) of open subsets of X which satis-�es the conditions (LO1) and (LO2) and induces tl contains some uncountablewell-ordered subchain, i.e. (O;�) or (O;�) is short which, in particular, meansthat tl is �rst countable (cf. [1]). In addition, the connectedness of t impliesthat none of the sets OnSO3O0$O O0 such that ? $ SO3O0$O O0 � O $ Xis empty. Let, therefore, S be a countable dense subset of X. Then wechoose for every point x 2 S some countable base of tl-neighborhoods of x.The union of the collection of these tl-neighborhoods with the countable setnO 2 Oj SO3O0$O O0 $ O ^SO3O0$O O0 \ XnO 6= ?o is a countable base oftl. Let us now assume that all linearly ordered subtopologies tl 2 TC are secondcountable. Then the same arguments that already have been applied in order togeneralize DT allow us to conclude that every continuous linear (total) preorder- on X has a continuous utility representation.On the other hand, it also cannot be expected that second countability ofthe linearly ordered subtopologies tl 2 TC of t is necessary in order to guaranteeusefulness of t. Indeed, let - be some arbitrary continuous linear preorderon X. Then the linearly ordered set (L;�) := (fL(x)gx2X ;�) satis�es thefollowing additional condition that strengthens condition (LO2).(LO3) : 8O 2 L( [L3O0$O O0 $ O =) [L3O0$O O0 � O):Therefore, it is somewhat surprising that in case that we concentrate onnormal topologies t on X (cf. De�nition 3.3) the conditions (LO1) and (LO2)completed by two straightforward conditions that are necessary in order to also 64 G. Herden, A. Pallackinclude the case that t is not necessarily connected already characterize usefultopologies. This characterization provides a generalization of EDT in the justdescribed way. In particular, it can be shown that our results are generalizationsof the theorem of Est�evez and Herv�es. The reader may still notice that the afore-discussed generalizations of DT and EDT provide a possibility of how to applyour results on useful topologies that will be proved in the following sections.3. R-separable systemsIt is well known that continuous linear preorders are closely related to R-separable systems (see [16]). Therefore, we shall approach the characterizationof useful topologies in a �rst step with help of R-separable systems.Suppose that R is an arbitrary binary relation on some �xed given topologi-cal space (X; t) (brie y we speak of an R-space). Then the reader may recall at�rst the following notation: A subset A of X is said to be R-decreasing (or sim-ply decreasing, if the relation is clear from the context), if a 2 A and bRa implythat b 2 A. An increasing set is de�ned in an analogous manner. Each subsetF of X gives rise to the smallest decreasing (respectively, increasing) subsetd(F) (respectively, i(F)) containing F . If F = fxg for some point x 2 X, thenwe write d(x) (respectively, i(x)) instead of d(fxg) (respectively, of i(fxg)). Foreach subset F of X there is a smallest closed decreasing subset D(F) (respec-tively, smallest closed increasing subset I(F)) containing F . If F = fxg forsome point x 2 X, then we write D(x) (respectively, I(x)) instead of D(fxg)(respectively, of I(fxg). Notice that for each subset F of X we have F � D(F).In general, this inequality is strict as is seen from the following simple example.Let X := f1;2g ; t := f?;f1g ;f2g ;f1;2gg and R := f(i;j) 2 X � X ji � jg.Then f2g = f2g $ D(2) = f1;2g = X. With these preliminaries we are fullyprepared for the following de�nition.De�nition 3.1. A family E of open R{decreasing subsets of X is said to be anR-separable system on X, if it satis�es the following conditions:(RS1) There exist sets E1;E2 2 E such that E1 � E2.(RS2) For all sets E1;E2 2 E such that E1 � E2 there exists some set E3 2 Esuch that E1 � E3 � E3 � E2.Moreover, if R is the equality relation \=" on X, i.e. the discrete order on X,we say that E is a separable system on X.Remark 3.2. In mathematical utility theory R-separable systems on X wereconstructed for the �rst time by Peleg [30] in order to prove his utility repre-sentation theorem. In Peleg`s Theorem R is a strict partial order or brie y anorder on X. In 1977 Burgess and Fitzpatrick [4] studied decreasing scales inX. We recall that a family S := fFrgr2D of open decreasing subsets of X issaid to be a decreasing scale in X, if the following two conditions are satis�ed:(DS1) D is a dense subset of the real interval [0;1] such that 1 2 D andF1 = X.(DS2) For every pair of real numbers r1 < r2 2 D the inclusion Fr1 � Fr2holds. Useful topologies and separable systems 65One immediately veri�es that decreasing scales in X are particular cases ofR-separable systems on X.The concept of an R-separable system on X is closely related to the conceptof a normally preordered space (cf. [29]) or more generally normal R-space.De�nition 3.3. An R-space (X; R; t) is said to be a normal R-space, if forany pair A, B of disjoint closed decreasing (respectively, increasing) subsets ofX there exist disjoint open decreasing, (respectively, increasing) subsets U, Vof X such that A � U and B � V .Notice that, if R coincides with the equality-relation " = " on X, then(X; R; t) is a normal space.The connections between the concept of an R{separable system and of anormal R{space is described in the following lemma ([16, Lemma 2.1]).Lemma 3.4. Let (X;R;t) be an arbitrary R-space. Then in order for (X;R;t)to be a normal R-space it is necessary and su�cient that for every pair C1, C2of disjoint closed subsets of X, C1 being decreasing and C2 increasing, thereexists an R-separable system E on X such that C1 � E and C2 � XnE forevery set E 2 E.Now we turn our attention to linear R-separable systems. Given an arbitraryR-space (X; R; t), an R-separable system E on X is said to be linear if, forevery pair of sets E, E0 2 E such that E 6= E0 at least one of the inclusionsE � E0 or E0 � E holds. Linear R-separable systems E on X easily can becharacterized ([17, Proposition 1.4.1]).Proposition 3.5. Let E be a family of open decreasing subsets of X, that islinearly ordered by set inclusion, and let B be the family of all sets E 2 E suchthat E $ E and for which there exists some set B 2 E such that E $ B. Thenthe following assertions are equivalent:(i) E is a linear R-separable system on X,(ii) 8E 2 B(TE$B2E B = TE$B2E B),(iii) 8E 2 B((E � TE$B2E B) ^ (TE$B2E B 2 E =) TE$B2E B =TE$B2E B)).Every R-separable system E on X contains some linear R-separable system.Indeed, let Q denote the rationals. Then this result is an immediate consequenceof the following lemma ([16, Lemma 2.2]).Lemma 3.6. Let E be an R-separable system on X. Then there exists a func-tion f : Q �! E such that f(p) � f(p) � f(q) for all p < q 2 Q.The reader may recall that a real-valued function f on X is said to be in-creasing if, for all pairs (x;y) 2 R, the inequality f(x) � f(y) holds. With helpof this notation we are already able to present the general separation theoremGST of Nachbin-Urysohn-type, which corresponds to GURT in [16] (see also[29], [34]). 66 G. Herden, A. PallackTheorem 3.7. Let (X; R; t) be an R-space. Then in order for (X; R; t) to bea normal R-space it is necessary and su�cient that for any two disjoint closedsubsets C1; C2 of X such that C1 is decreasing and C2 increasing, there existssome continuous increasing real-valued function f on X such that 0 � f � 1,f(x) = 0 for all x 2 C1 and f(x) = 1 for all x 2 C2.Now we present the most important result of this section (see [16, Lemma2.3]).Theorem 3.8. Let (X; R; t) be an R-space. Then every linear R-separablesystem E on X induces a linear preorder - on X which satis�es the followingproperties:(L1): R �-;,(L2): The order topology t- is coarser than tE.For later use we recall the de�nition of -. Let E be a linear R-separable sys-tem on X and let points x, y 2 X be arbitrarily chosen. Set Ey := fE 2 Ejy 2 Egand de�ne - by settingx - y () Ey = ? _ 8E 2 Ey8B 2 Ey(E 6� B _ x 2 B):It is easily seen that - can be divided into the following two less complicatedsubrelations:(i) x � y () there exists some R-separable systemB � E on X such thatx 2 B and y 2 XnB for all sets B 2 B;(ii) x � y () :(x � y) and :(y � x):It seems that Theorem 3.8 is closely related to the famous Szpilrajn's Theo-rem [33] which states that every partially order can be re�ned to a linear order.But - is not necessarily a re�nement of A, since we did not require that � iscontained in the set of all pairs (x;y) 2 R such that (y;x) 62 R. For later usewe abbreviate this set by RS. Hence, Szpilrajn's Theorem is not a consequenceof Theorem 3.8. As far as the authors know the continuous analogous of Szpil-rajn's Theorem never has been discussed in the literature. In order to be moreprecise, the reader may recall that a linear preorder - on X is continuous if andonly if for each pair of points x � y 2 X there exists some continuous increasingreal-valued function fxy on X such that fxy(x) < fxy(y) (the reader may apply[22, Lemma 1] and GST to verify this result). Obviously, this characterizationof continuous linear preorders can be generalized to arbitrary preorders. There-fore, a continuous preorder - on X is said to satisfy the Szpilrajn-property, ifthere exists a continuous linear preorder -� on X such that -� is a re�nementof -. The Szpilrajn-property will be discussed in [17, Chapter 6].4. Real order-embeddingsLet throughout this section X be some �xed given set and R some relationon X. We recall some de�nitions. (X; R) is said to be Ja�ray-separable ifthere exists a countable subset Z of X such that, if x;y 2 X and (x;y) 2 RS,then there exist points z;z0 2 Z such that xRxRSz0Ry. (X; R) is said to beBirkho�-separable if there exists a countable subset Z of X such that, for every Useful topologies and separable systems 67pair (x;y) 2 RS \(XnZ)�(XnZ), there exists some z 2 Z such that xRSzRSy.The space (X; R) is called Debreu-separable if there exists a countable subsetZ of X such that for every pair (x;y) 2 RS there exists some z 2 Z such thatxRzRy, and it is called Cantor-separable if there exists a countable subset Zof X such that, for every pair (x;y) 2 RS, there exists some z 2 Z such thatxRSzRSy.Now we are fully prepared for presenting the following Representation The-orem (see, for example, [3, Proposition 1.6.11] and [13, Lemma 3.1]).Theorem 4.1. Let (X; -) be a linearly preordered set. Then the followingassertions are equivalent:(i) There exists an order preserving function f : (X; -) �! (R; �),(ii) (X; -) is Ja�ray-separable,(iii) (Xj�; -j�) is Birkho�-separable,(iv) (X; -) is Debreu-separable,(v) (X; t-) is separable and (Xj�; -j�) has only countably many jumps,(vi) (X; t-) is second countable.The reader may notice that in contrast to Proposition 1.6.11 in Bridgesand Mehta [3] the assertion concerning Birkho�-separability has been mod-i�ed somewhat. Indeed, the concepts of Birkho�-separability and Debreu-separability are not equivalent in the context of preorders but only in the contextof orders. This hint is due to Mehta [24, November 1999, oral communication].5. The structure of useful topologiesLet X be a �xed given set and let t be some topology on X. It is the aimof this section to characterize all useful topologies on X with help of linearseparable systems on X. Because of Proposition 3.5 the characterization ofuseful topologies with help of linear separable systems is a quite satisfactoryapproximation of the desired results that have been announced in the secondsection. In order to also include the non-connected case we need at �rst thefollowing notation. A topological space (X; t) is said to satisfy the open-closedcountable chain condition (OCCC), if every family F of non-empty open andclosed subsets F of X that satis�es the following two conditions is countable:(OC1): 8F 2 F(F � F 0 _ F 0 � F),(OC2): 8F 2 F([fF 0 2 F jF 0 $ Fg $ F $ \fF 00 2 F jF $ F 00g).Let now E be a linear separable system on X. We consider the set Z(E)of all pairs B $ E 2 E for which there exists some set C 2 E such thatB $ C � C $ E. Then E is said to have a countable re�nement if there existsa countable family O of non{empty open subsets of X such that for every pair(B;E) 2 Z(E) there exists some set O 2 O such that O � E \ XnB. E issaid to be second countable, if there exists a countable subset H of E such thatfor every pair of sets (B;E) 2 Z(E) there exists some set E+ 2 H such thatB � E+ � E+ � E. In addition, if G(E) is the set of all (open) sets E 2 E suchthat SE3B$E B $ E, let GG denote the family of all linear separable systems E 68 G. Herden, A. Pallackon X for which G(E) is a countable set. With help of this notation the followingproposition characterizes all useful topologies t on X.Proposition 5.1. For a topology t on a set X, the following assertions areequivalent:(i) t is useful,(ii) t satis�es OCCC and every linear separable system E on X has acountable re�nement,(iii) t satis�es OCCC and every linear separable system E on X is secondcountable,(iv) t satis�es OCCC and every linearly ordered subtopology tl of t that isinduced by some linear separable system E 2 GG is second countable.Proof. (i) =) (ii) At �rst we assume, in contrast, that t does not satisfy OCCC.Then there exists an uncountable family F of non-empty open and closed sub-sets F of X that satis�es the conditions (OC1) and (OC2). Since every setF 2 F is open and closed, condition (OC1) implies that the preorder - de�nedfor every pair of points x;y 2 X byx - y () 8F 2 F(y 2 F =) x 2 F)is linear and continuous. In addition, the uncountability of F allows us toconclude with help of condition (OC2) that - has uncountably many jumps.Indeed, for every set F 2 F any pair of points x 2 Fn [ fB 2 F jB $ Fg,y 2 \fC 2 F jF $ CgnF de�nes a jump of -. Hence, - is not representable.This contradiction implies that t satis�es OCCC. Let now E be a linear separablesystem on X. It remains to show that there exists a countable family O ofopen subsets of X such that for every pair (B;E) 2 Z(E) there exists some setO 2 O such that O � E \XnB. As remarked after Theorem 3.8 we may de�nea continuous linear preorder - on X induced by E such that for every pair(B;E) 2 Z(E) and every pair of points x 2 CnB;y 2 EnC the strict inequalityx � y holds. This means that we may choose for every pair (B;E) 2 Z(E)points x � y 2 X such that ]x;y[� E \ XnB. Because of assertion (i), thelinear preorder - is representable. This means, in particular, that (Xj�; -j�)only has countably many jumps and that t- is second countable (cf. Theorem4.1, assertions (v) and (vi)). The existence of the desired family O of opensubsets of X, thus, follows immediately, which �nishes the proof of assertion(ii).(ii) =) (i) Let - be a continuous linear preorder on X. Because of the OpenGap Lemma ([9, 10]) it su�ces to prove that - is representable. Therefore, weconsider the linear separable system L := fL(x)gx2X := ffy 2 X jy � xggx2Xon X. Since (X; t) satis�es OCCC it follows that (Xj�; -j�) only has countablymany jumps. Indeed, let f([xi]; [yi])gi2I be the family of all jumps of (Xj�; -j�). Then we may choose for every index i 2 I the open and closed subsetFi := fz 2 X jz - xig = fz 2 X jz � yig of X. Let F be the family of thesesubsets. Because (X; -) is a chain, we may conclude that F satis�es condition(OC1). In addition, the de�nition of F implies that F also satis�es condition Useful topologies and separable systems 69(OC2). Hence, it follows from OCCC that F is countable, and this means thatthe family f([xi]; [yi])gi2I of all jumps of (Xj�; -j�) actually is countable. Inorder to now �nish the proof of the representability of - it remains to verifythat (X; t-) is separable (cf. Theorem 4.1 (v)). L has a countable re�nement.Hence, there exists a countable family O of non{empty open subsets of X suchthat for every pair x � y 2 X for which ]x;y[ is neither empty nor contains ajump of (Xj�; -j�), there exists some set O 2 O such that O �]x;y[. Choosingin every set O 2 O some point x 2 O and considering, in addition, for everyjump ([x]; [y]) of (Xj�; -j�) points x 2 [x] and y 2 [y] respectively, we mayconclude that (X; t-) must be separable, and assertion (i) follows.(i)^(ii) =) (iii) Let E be a linear separable system on X. It su�ces to showthat E is second countable. Let, therefore, O be a countable family of non{empty open subsets of X such that for every pair of sets (B;E) 2 Z(E) thereexists some set O 2 O such that O � E \XnB. By eliminating redundant setswe may assume without loss of generality that for every set O 2 O there existsets B � E 2 E such that O � E \ XnB. Hence, we may choose for every setO 2 O the non{empty linear separable systems W1 := fB 2 E jOnB 6= ?g andW2 := fE 2 E jOnE = ?g. It follows that there exist countable sets O1 � W1and O2 � W2 such that SE2O1 E = SE2W1 E and TE2O2 E = TE2W2 E.Indeed, otherwise the construction described after Theorem 3.8 implies thatboth continuous linear preorders -1 and -2 on X which are induced by W1and by W2, respectively, are not short and, thus, not representable in contrastto assertion (i). Since O is countable we may conclude that B := SO2O O1 [SO2O O2 is a countable set. The construction of B implies that for every pair ofsets (B;E) 2 Z(E) there exists some set E+ 2 B such that B � E+ � E+ � E,as desired.(iii) =) (iv) Trivial.(iv) =) (i) Let - be some continuous linear (total) preorder on X. Thenwe consider the linear separable system L := fL(x)gx2X on X. In the proofof the implication (ii) =) (i) it already has been shown that OCCC impliesthat (Xj�; -j�) only has countably many jumps. Since L satis�es condition(LO3) it follows that L 2 GG. The reader may recall that condition (LO3)implies condition (LO2). Assertion (iv), thus, implies that the linearly orderedsubtopology tl of t that is induced by L is second countable which allows usto conclude with help of the considerations in the second section that - hasa continuous utility representation. Therefore, the proof of the proposition iscomplete. �Clearly, in case that t is connected OCCC may be omitted. Hence, thecharacterization of t to be useful simpli�es somewhat.Corollary 5.2. Let t be connected. Then the following assertions are equiva-lent:(i) t is useful,(ii) every linear separable system E on X has a countable re�nement,(iii) every linear separable system E on X is second countable, 70 G. Herden, A. Pallack(iv) every linear ordered subtopology tl of t that is induced by some linearseparable system E 2 GG is second countable.6. A different approachLet (X; t) be an arbitrary topological space and let G := fEigi2I be a familyof separable systems on X. Then G is said to be well-separated, if it satis�esthe following conditions:(WS1): 8i 2 I 8j 2 I 8E 2 Ei8B 2 Ej(i 6= j =) E \ B = ?).(WS2): 8fEigi2I (Ei 2 Ei =) Si2I Ei = Si2I Ei).Now the following lemma holds:Lemma 6.1. Let t be a useful topology on X. Then every well-separated familyG := fEigi2I of separable systems on X is countable.Proof. Let G := fEigi2I be some well-separated family of separable systems onX. Then we may assume without loss of generality that every subset J of I forwhich there exists for every j 2 J some non{empty open and closed set Ej 2 Ejis countable. Indeed, otherwise we consider some well ordering � on J, choosefor every j 2 J some �xed non{empty open and closed set Ej 2 Ej in orderto consider for every j 2 J the set Fj := Si�j Ei. Then the conditions (WS1)and (WS2) imply that F := fFjgj2J is an uncountable family of non{emptyopen and closed subsets of X that satis�es the conditions (OC1) and (OC2)and, thus, contradicts the usefulness of t.Let us now assume, in contrast, that I is uncountable. Then we consider somewell-ordering � on I, choose the �rst uncountable ordinal !1 and consider somesubfamily fE�g� 0 and someuncountable subset S of Xj� such that �(x;y) � � for all points x 6= y 2 S.Indeed, otherwise, for every natural number n > 0, every subset Zn of Xj� suchthat �(x;y) � 1n for all points x 6= y 2 Zn is countable. The Lemma of Zornallows us to choose, for every natural number n > 0, some maximal subset Ynof Xj� such that �(x;y) � 1n for all points x 6= y 2 Yn. Then Y := Sn2Nnf0g Ynis a countable subset of Xj� such that Y = Xj�, a contradiction. Thus, theexistence of S follows. The inclusion t� � t implies that t�j� � tj�. Hence, wemay conclude that the family ��y 2 Xj� j�(x;y) < �3 x2S is an uncountablefamily of pairwise disjoint non{empty open subsets of Xj� that, obviously, islocally �nite. This contradiction �nishes the proof. �Now we are ready to summarize our considerations for some interesting re-sults.Proposition 6.5. Let �(X;C(X)) be induced by some uniformity that has acountable base. Then the following assertions are equivalent:(i) t is useful,(ii) �(X; C(X)) is second countable. Useful topologies and separable systems 73Proof. (i) =) (ii) Since �(X;C(X)) is induced by some uniformity which hasa countable base we may conclude that �(X;C(X)) is induced by some pseu-dometric � on X. With help of the inclusion �(X;C(X)) � t and Lemma 6.3the desired conclusion now follows from Lemma 6.4.(ii) =) (i) Let - be a continuous linear preorder on X. Then t- ��(X;C(X)) and assertion (i) follows from DT. �Corollary 6.6. Let (G; �; t) be a �rst countable topological group. Then thefollowing assertions are equivalent:(i) t is useful,(ii) t is second countable.Proof. Every topology t of a �rst countable topological group (G; �; t) is inducedby some uniformity which has a countable base (see, for example, [18]). �Corollary 6.7. [13] Let t be induced by some metric �. Then the followingassertions are equivalent:(i) t is useful,(ii) t is second countable.Remark 6.8. Corollary 6.7 has an important consequence. It implies, in par-ticular, that for a metric space (X;�) the assumptions of Debreu's Theorem arenot only su�cient but also necessary for a continuous linear preorder - on (X;�)to be representable by a continuous utility function. On the other hand, metricspaces (in particular Hilbert spaces or more generally Banach spaces) which arenot second countable are meanwhile commonly encountered in economic theory(see [23] or our remark in the introduction). This is the case, for example, if thecommodity space is L1(�), the space of �-essentially bounded �-measurablefunctions on a �-�nite measure space, which arises in the analysis of allocationof resources over time or states of nature ([2]), or ca(K), the space of countablyadditive signed measures on a compact metric space which has been exploitedfor the analysis of commodity di�erentiation ([21] and [19]). Linear preordersde�ned on these spaces principally must satisfy more properties than just be-ing continuous in order to have a continuous utility representation. Hence, theapproaches of Shafer [31], Mas-Colell [21], Monteiro [28], Mehta and Monteiro[25] and others gain additional importance. The problem which arises is tolook for useful natural additional conditions which a linear preorder on thesespaces should satisfy and which also guarantee its representability by a contin-uous utility function. Such a useful condition could be countably boundedness.The reader may recall that a linear preorder - on X is countably bounded ifthere exists a countable subset Y of X such that for every point x 2 X thereexist points y;y0 2 Y such that y - x - y0. For example, since every convexsubset of the space L1(�) and ca(K) respectively is path connected, it followsfrom Monteiro [28] that every continuous countably bounded linear preorderon a convex subset of L1(�) and ca(K) respectively has a continuous utilityrepresentation. Another useful condition could be convexity (see [5, Theorem3]). 74 G. Herden, A. PallackIn order to now prove the main result of this section let t be an arbitrarybut �xed given topology on X. In case that t0 is a topology on X that cannotbe excluded to be di�erent from t we denote for every subset A of X by c0(A)the t0-closure of A. In case that we are sure that t0 = t the t-closure of Ais abbreviated as usual by A. Then a topology t0 on X is said to be well-compatible, if for every t0-open subset O of X and every point x 2 O thet0-closure c0(fxg) of fxg is contained in O. The reader may recall that t0 iswell-compatible, if and only if for each pair of points x;y 2 X the equivalencec0(fxg) = c0(fyg) () c0(fxg) \ c0(fyg) 6= ? holds. Now Lt is the family of allwell-compatible topologies t0 on X for which there exists some linear separablesystem E 2 GG (cf. section 5) such that E � t0 � tE and c0(E0) � E for everypair of sets E0 � E0 � E 2 E. tE is the topology on X that is induced by E(cf. section 3). The reader may prove as an easy exercise that c0(E0) � E forevery pair of sets E0 � E0 $ E 2 E, if and only if (c0(E)nE) \ c0(E0) = ? forevery pair of sets E0 � E0 $ E 2 E . In case that t is connected it follows thatLt is the set of all well-compatible topologies t0 on X for which there existssome linear separable system E on X such that E � t0 � tE and c0(E0) � E forevery pair of sets E0 � E0 $ E 2 E. Let �nally E be some arbitrarily chosenlinear separable system on X. Then the reader may verify, in addition, thattE 2 Lt, if and only if for every pair of sets E0 � E0 � E 2 E the equationE0 = \�E00 2 E jE0 � E00 � E holds and, furthermore, a possible �rst elementor a possible last element of the chain (E;�) is open and closed.Now we are fully prepared for proving the main result of this section.Proposition 6.9. The following assertions are equivalent:(i) t is useful,(ii) t satis�es OCCC, every well-separated family G := fEigi2I of separablesystems on X is countable and every topology t0 2 Lt is pseudometriz-able.(iii) t satis�es OCCC, �(X;C(X)) satis�es CLF and every topology t0 2 Ltis pseudometrizable.Proof. (i) =) (ii) Because of Proposition 5.1 and Lemma 6.1 it is su�cientto prove that every topology t0 2 Lt is pseudometrizable. Let, therefore,some topology t0 2 Lt be arbitrarily chosen. Then there exists some linearseparable system E on X, whose associated set G(E) is countable, such thatE � t0 � tE and c0(E0) � E for every pair of sets E0 � E0 � E 2 E. Letnow t0j� be the quotient topology that is induced by the equivalence relation\x � y () c0(fxg) = c0(fyg)\. The well-compatibility of t0 implies that theequivalence relation � is open, i.e. the canonical projection p : X �! Xj�is open. This means, in particular, that t0 is pseudometrizable, if and onlyif t0j� is metrizable. In order to verify that t0j� is metrizable we show at �rstthat t0j� is second countable. Then we prove that t0j� is normal which �nallyallows us to conclude with help of the Alexandro�-Urysohn Metrization The-orem that t0j� is metrizable. One more application of the well compatibility Useful topologies and separable systems 75of t0 implies that t0j� is second countable, if and only if t0 is second count-able. Since G(E) is countable it follows with help of Proposition 5.1 (iii)that tE is second countable. This means that there exists a countable setC(E) of pairs of sets E0 � E0 � E 2 E [ f?;Xg such that the family ofcorresponding open sets E \ XnE0 is a base of tE. We want to show thatB := �E \ Xnc0(E0) jE0 � E0 � E 2 C(E) is a base of t0. Then the secondcountability of t0 follows. Let, therefore, O be some (non-empty) t0-open subsetof X and let x 2 O be some arbitrary point. We must show that there existssome set E+ \ Xnc0(E++) 2 B such that x 2 E+ \ Xnc0(E++) � O. Theinclusion t0 � tE implies that there exists some pair of sets E0 � E0 $ E 2 C(E)such that x 2 E \ XnE0 � O. We, thus, distinguish between the following twocases:Case 1. x 2 Enc0(E0). In this case the inclusion E0 � c0(E0) implies thatx 2 E \ Xnc0(E0) � E \ XnE0 � O and we are done.Case 2. x 62 Enc0(E0), i.e. x 2 c0(E0)nE0 � c0(E0)nE0. In this situation weshow at �rst that E0nE0 \O 6= ?. In order to verify this inequality the validityof the equation c0(E0nE0) = c0(E0)nE0 is needed. Since E0nE0 � c0(E0)nE0 andc0(E0)nE0 is t0-closed the inclusion c0(E0nE0) � c0(E0)nE0 follows immediately.Hence, the desired equation will be proved if we are able to show that also theinclusion c0(E0)nE0 � c0(E0nE0) holds. Let, therefore, some point y 2 c0(E0)nE0and some t0-neighborhood U of y be arbitrarily chosen. We must show thatU \ E0nE0 6= ?. Then y 2 c0(E0nE0) and the inclusion follows. The inclusiont0 � tE implies that there exist some pair of sets E�� � E�� $ E� 2 C(E) suchthat y 2 E� \ XnE�� � U. Since y 2 E�nE0 the linearity of E implies thatE0 � E�. On the other hand, it follows that E�� � E0. Indeed, otherwise wemay conclude that E0 $ E��, which means that c0(E0) � E��. Since y 2 c0(E0)this inclusion contradicts the relation y 2 E \ XnE��. A combination of theseconsiderations implies that E0nE0 � E \XnE��. Hence, U \E0nE0 6= ?, which�nishes the proof of the desired equation. With help of the equation c0(E0nE0) =c0(E0)nE0 the inequality E0nE0\O 6= ? easily can be veri�ed. Indeed, otherwisewe may conclude that E0nE0 � XnO. Since XnO is t0-closed it, thus, followsthat c0(E0)nE0 = c0(E0nE0) � XnO which contradicts the relation x 2 c0(E0)nE0and x 2 O. Now we proceed by choosing some arbitrary point z 2 E0nE0 \ Oin order to then consider some pair of sets E++ � E++ $ E+ 2 C(E) suchthat z 2 E+ \ XnE++ � O. Since z 2 E0nE0 and, thus, z 2 E+nE0 thelinearity of E implies that E0 � E0 � E+ which, in particular, means thatx 2 c0(E0) � E+. Because of the relations z 62 E++ and z 2 E0 it follows, onthe other hand, that E++ � E++ $ E0. Hence, c0(E++) � E0. We, thus, maysummarize our considerations for the conclusions x 2 E+\Xnc0(E++) � O andE+ \ Xnc0(E++) 2 B which completes the second case and, therefore, showsthat t0 is second countable. The particular construction of B �nally allows usto apply the arguments of the proof of Proposition 1.4.2 in [17] in order toverify that t0 is normal. Then the well-compatibility of t0 implies that also t0j� 76 G. Herden, A. Pallackis normal and the Alexandro�-Urysohn Metrization Theorem can be applied.This last conclusion �nishes the proof of assertion (ii).(ii) =) (iii) This implication follows immediately with help of Corollary 6.2.(iii) =) (i) Let - be some arbitrary continuous linear (total) preorderon X. Then we consider the linear separable system L := fL(X)gx2X :=ffy 2 X jy � xggx2X on X. Since t satis�es OCCC it follows that - only hascountably many jumps which means that G(L) is countable (cf. the corre-sponding argument in the proof of the implication (ii) =) (i) of Proposition5.1. Clearly, - coincides with the linear preorder on X that is induced by L(cf. Theorem 3.8). Hence, we may apply Theorem 3.8 in order to concludethat the order topology t- that is induced by - is coarser than tL. For everypoint x 2 X its t--closure c0(fxg) coincides with the equivalence class [x] thatis de�ned by �. Hence, it follows that t- is well{compatible. Since, in addition,L � t- and d(x) = fy 2 X jy - xg � L(z) = fu 2 X ju � zg for every pair ofpoints y � z 2 X we may conclude that t- 2 Lt. Hence, t- is pseudometriz-able. The underlying argument which the proof of Lemma 6.3 is based uponimplies that t- � �(X;C(X)). This means, in particular, that t- satis�es CLF.Therefore, it follows from Lemma 6.4 that t- is second countable which impliesthat - has a continuous utility representation. This last conclusion settles theimplication (iii) =) (i) and nothing remains to be shown. �Corollary 6.10. Let t be connected. Then the following assertions are equiv-alent:(i) t is useful,(ii) Every well-separated family G := fEigi2I of separable systems on X iscountable and every topology t0 2 Lt is pseudometrizable,(iii) �(X;C(X)) satis�es CLF and every topology t0 2 Lt is pseudometriz-able.Remark 6.11. The condition that every topology t0 2 Lt is pseudometrizableseems to be a bit arti�cial. On the other hand, Proposition 6.9 means that t isuseful if and only if t satis�es OCCC, �(X;C(X)) satis�es CLF and t allows thede�nition of enough (continuous) pseudometrics on X. Hence, Proposition 6.9which, in particular, generalizes the nice result of Est�evez and Herv�es completesProposition 5.1 and may at least serve as basis for �nally obtaining still moresatisfactory results. 7. Useful normal topologiesIn the second section we already have announced some optimal result on theusefulness of normal topologies. In order to prove this result let t be a �xedgiven normal topology on X. For every subset A of X the interior of A isdenoted by A�. Then we choose the family O of all sets O of open subsets Oof X that are linearly ordered by set inclusion and satisfy condition (LO1) (cf.section 5).Let some set O 2 O be arbitrarily chosen. Then the sets Z(O), G(O) and OGand the concepts of O to have a countable re�nement or to be second countable Useful topologies and separable systems 77are de�ned in the same way as the corresponding sets Z(O), G(O) and GG, andthe similar concepts in section 5. In addition, we consider the family F of allsets O 2 O which also satisfy the following condition which completes condition(LO2) in order to also include the case that t is not necessarily connected (cf.section 2).LO2+ : ������8<:O 2 O j [O3O0$O O0 $ O ^ [O3O0$O O0 \ XnO 6= ?9=;������++������8<:O 2 O j [O3O0$O O0 � O ^ [O3O0$O O0 = ( [O3O0$O O0)� _ O = O9=;������ �� ! :The reader may verify that in case that t is connected the conditions (LO2)and (LO2+) coincide.Proposition 7.1. Let t be a normal topology on X. Then the following asser-tions are equivalent:(i) t is useful,(ii) t satis�es OCCC and every set O 2 O has a countable re�nement,(iii) t satis�es OCCC and every set O 2 O is second countable,(iv) t satis�es OCCC and every linearly ordered subtopology tl of t that isinduced by some set O 2 OG is second countable,(v) t satis�es OCCC and every linearly ordered subtopology tl of t that isinduced by some set O 2 F is second countable.Proof. (i) =) (ii) Let some set O 2 O be arbitrarily chosen. Then we considerthe set M(O) of all sets O 2 O for which there exists some maximal setO 3 O0 $ O. Since t is a normal topology on X it follows from Lemma3.4 and Lemma 3.6 with help of condition (LO1) that for every pair of setsO 3 O0 $ O 2 M(O) there exists a linear separable system E(O) on X suchthat O0 � E � O for every set E 2 E(O). We, thus, set E(O) := O [(SO2M(O) E(O)), and show that E(O) is a linear separable system on X. Let,therefore, E0(O) � E(O) be the subset of all sets E 2 E(O) such that E $ E andfor which U(E) := fE0 2 E(O) jE $ E0g 6= ?. Then we choose some arbitraryset E 2 E0(O) and distinguish between the following two cases:Case 1: (U(E);�) does not contain a minimal element. In this case theconstruction of E(O) allows us to conclude with help of condition (LO1) thatTE02U(E) E0 = TE02U(E) E0.Case 2: (U(E);�) contains a minimal element. Let E0 be this minimalelement of (U(E);�). Then the de�nition of M(O) implies with help of theconstruction of E(O) that E0 is closed.Summarizing both cases it follows with help of Proposition 3.5 (ii) that E(O),actually, is a linear separable system on X. Assertion (ii) now is an immediateconsequence of the corresponding assertion of Proposition 5.1. 78 G. Herden, A. Pallack(ii) =) (i) Since every linear separable system E on X satis�es condition(LO1) the desired implication follows with help of the implication (ii) =) (i)in the proof of Proposition 5.1.(i) =) (iii) Let O 2 O be some arbitrarily chosen set. As in the proofof the implication (i) =) (ii) we consider the set M(O) and construct thelinear separable system E(O) on X. Of course, we may assume without loss ofgenerality that for every pair of sets O 3 O0 $ O 2 M(O) such that O0 or Ois closed the corresponding linear separable system E(O) on X consists of O0and O. Let us abbreviate this assumption by (*). Because of Proposition 5.1(iii) there exists some countable subset E0 of E0(O) such that for every pair ofsets (E0;E) 2 Z(E(O)) there exists some set E+ 2 E0 such that E0 � E+ �E+ � E. Now we consider the situation O0 $ O00 � O00 $ O for some pairof sets (O0;O) 2 Z(O) and some set O00 2 O. There exists some set E 2 E0such that O0 � E � E � O. If E 62 O, then there exists because of (*) andthe construction of E(O) some pair of sets O 3 O+ $ O++ 2 M(O) suchthat O+ $ E � E $ O++. Since O is linearly ordered by set inclusion itfollows with help of condition (LO1) and the chain O0 $ O00 � O00 $ O thatO0 $ O+ � O+ � O++ $ O. Hence, one immediately veri�es that assertion(iii) will follow with help of assertion (iii) of Proposition 5.1, if we are able toshow that the set of all pairs O 3 O+ $ O++ 2 M(O) for which there existssome set E 2 E0nO such that O+ � E � E � O++ 2 M(O) is countable.But this is easily seen since the corresponding sets O++ \ XnO+ are pairwisedisjoint and E0 is countable.(iii) =) (iv) Trivial.(iv) =) (i) In the same way as the implication (ii) =) (i) also this implica-tion follows with help of the proof of the corresponding implication (iv) =) (i)of Proposition 5.1.(i) ^ (iv) =) (v) Let some set O 2 F be arbitrarily chosen. Then weconsider the linear separable system E(O) on X that already has been con-structed in the proof of the implication (i) =) (ii). Since the linearly orderedsubtopology tl of t that is induced by O is coarser than the linearly orderedsubtopology t0l of t that is induced by E(O) it su�ces to verify that t0l is sec-ond countable (cf. the argument of the generalization of DT in the secondsection). In order to show that t0l is second countable it is because of assertion(iv) and condition (LO2+) su�cient to prove that the set K(O) of all (open)sets O 2 O such that (SO3O0$O O0)� $ SO3O0$O O0 $ O $ O is countable.In order to show the countability of K(O) we apply the normality of t in or-der to construct for every (open) set O 2 K(O) some linear separable systemE0(O) on X such that SO3O0$O O0 � E0 � E0 � O for every set E0 2 E0(O)(cf. corresponding argument in the proof of the implication (i) =) (ii)). Since(SO3O0$O O0)� $ SO3O0$O O0 $ O $ O we may conclude that E0(O) 6= ?.In the same way as in the corresponding part of the proof of the implication(i) =) (ii) it follows that E0(O) := E(O) [ (SO2K(O) E0(O)) also is a linearseparable system on X. Let us now assume, in contrast, that K(O) is not Useful topologies and separable systems 79countable. Then, since E0(O) 6= ? for every (open) set O 2 K(O) we mayconclude that the continuous linear (total) preorder - on X that is inducedby E0(O) has uncountably many jumps or contains an uncountable family ofpairwise disjoint open (non-degenerate) intervals. This means, in particular,that - has no continuous utility representation, which contradicts assertion (i).Hence, assertion (v) follows.(v) =) (iv) Since OG � F assertion (iv) is an immediate consequence ofassertion (v). �In case that t is connected, Proposition 7.1 is the generalization of EDT tonormal topologies (cf. the corresponding remark in section 2).Corollary 7.2. Let t be a normal and connected topology on X. Then thefollowing assertions are equivalent:(i) t is useful,(ii) every set O 2 O has a countable re�nement,(iii) every set O 2 O is second countable,(iv) every linearly ordered subtopology tl of t that is induced by some setO 2 OG is second countable,(v) every linearly ordered subtopology tl of t that is induced by some setO 2 F is second countable.Let, for the moment, a normal topology t on X said to be short, if every setO 2 O that is well-ordered by set inclusion is countable. Then the followinginteresting proposition holds which, in particular, shows that condition (LO1)is a generalization of CLF. This means that Proposition 7.1 and Corollary 7.2are generalizations of the theorem of Est�evez and Herv�es.Proposition 7.3. In order for a normal topology t on X to be short it isnecessary that t satis�es CLF.Proof. Let t be short. We assume, in contrast, that t does not satisfy CLF.Then there exists an uncountable locally �nite family O := fOigi2I of pairwisedisjoint (non-empty) open subsets of X. In analogy to the proof of Lemma6.1 we may assume that none of the sets Oi (i 2 I) contains some non-emptyopen and closed subset. Let us abbreviate this assumption by (*). In addition,the proof of Lemma 6.1 allows us to assume that I coincides with the �rstuncountable ordinal !1, i.e., O = fOigi2I = fO�g�