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Applied General TopologyUniversidad Polit�ecnica de ValenciaVolume 1, No. 1, 2000pp. 93 - 98

Separation axioms in topological preorderedspaces and the existence of continuousorder-preserving functionsG. Bosi, R. IslerAbstract. We characterize the existence of a real continu-ous order-preserving function on a topological preordered space,under the hypotheses that the topological space is normal andthe preorder satis�es a strong continuity assumption, called IC-continuity. Under the same continuity assumption concerning thepreorder, we present a su�cient condition for the existence of acontinuous order-preserving function in case that the topologicalspace is completely regular.2000 AMS Classi�cation: 06A06, 54F30Keywords: topological preordered space, decreasing scale, order-preservingfunction 1. IntroductionMcCartan [?] introduced a natural continuity hypothesis on a preorder � ona topological space (X; �). Such an assumption, which is called IC-continuitythroughout this paper, is stronger than the usual hypothesis according to whichall the lower and upper sections are closed. Separation axioms in ordered topo-logical spaces were studied, in connection with suitable continuity assumptions,by Burgess and Fitzpatrick [?] and later by K�unzi [?]. On the other hand, theexistence of a real continuous order-preserving function on a normally pre-ordered topological space was characterized by Mehta [?].In this paper we are concerned with the existence of a real continuous order-preserving function f on a topological preordered space (X; �; �) in case that(X; �) is either a normal or a completely regular space, and the preorder � isIC-continuous. Such a problem was already faced by Bosi and Isler [?]. Werecall that a full characterization of the existence of a real continuous order-preserving function on a topological preordered space was provided by Herden[?], [?] (see also Mehta [?] for an excellent review), who introduced the notion



94 G. Bosi, R. Islerof a separable system. Such a concept appears as a generalization of the conceptof a decreasing scale, which was used by Burgess and Fitzpatrick [?] and seemsmore suitable to our aims.2. Definitions and preliminary considerationsGiven a preorder � on an arbitrary set X (i.e., a binary relation on X whichis re
exive and transitive), denote by � and � the asymmetric part and re-spectively the symmetric part of �. If (X; �) is a preordered set, and � is atopology on X, then the triplet (X; �; �) will be referred to as a topologicalpreordered space.A subset A of a set X endowed with a preorder � is said to be decreasing(respectively, increasing) if (x 2 A) ^ (y � x) ) y 2 A (respectively, (x 2A) ^ (x � y) ) y 2 A).If A is any subset of a set X endowed with a preorder �, then denote byd(A) (respectively by i(A)) the intersection of all the decreasing (respectively,increasing) subsets of X containing A.Given a topological preordered space (X; �; �), we say that � is(i) continuous if d(x) = d(fxg) and i(x) = i(fxg) are closed sets for everyx 2 X,(ii) I-continuous if d(A) and i(A) are open sets for every open subset A ofX,(iii) C-continuous if d(A) and i(A) are closed sets for every closed subset Aof X,(iv) IC-continuous if � is both I-continuous and C-continuous.De�nitions (ii) and (iii) were introduced by McCartan [?]. The previousterminology is similar to the terminology adopted by K�unzi [?]. Obviously,given a topological preordered space (X;�;�), if (X;�) is a T1 space, and � isC-continuous, then � is continuous. So, if the T1 separation axiom holds, theconcept of C-continuity is stronger than the concept of continuity of a preorderon a topological space.From Nachbin [?], a topological preordered space (X;�;�) is said to be nor-mally preordered if, given a closed decreasing set F0 and a closed increasing setF1 with F0 \F1 = ?, there exist an open decreasing set A0 containing F0, andan open increasing set A1 containing F1 such that A0 \ A1 = ?.It is easily seen that a topological preordered space (X;�;�) is normallypreordered if (X;�) is normal and � is IC-continuous. Indeed, given a closeddecreasing set F0 and a closed increasing set F1 with F0\F1 = ?, from normalityof (X;�) there exist an open set A00 containing F0, and an open set A01 containingF1 such that A00 \ A01 = ?, and from IC-continuity of the preorder � we havethat A0 = d(A00)ni�A01 n d(A00)� is an open decreasing set containing F0, A1 =i(A01)nd�A00 n i(A01)� is an open increasing set containing F1, and A0\A1 = ?.From Herden [?], a topological preordered space (X;�;�) is Nachbin sepa-rable if there exists a countable family fAn;Bngn2N of pairs of closed disjoint



Separation axioms in topological preordered spaces and... 95subsets of X such that An is decreasing, Bn is increasing, and f(x;y) 2 X �X :x � yg � Sn2N (An � Bn).From Burgess and Fitzpatrick [?], given a topological preordered space (X,�, �), a family G = fGr : r 2 Sg of open decreasing subsets of X is said to bea decreasing scale in (X;�;�) if the following conditions are satis�ed:(i) S is a dense subset of [0; 1] such that 1 2 S and G1 = X, and(ii) for every r1;r2 2 S with r1 < r2, it is Gr1 � Gr2.Observe that any decreasing scale G in a topological preordered space (X;�;�)is a linear separable system in Herden's terminology (see Herden [?]).If (X;�) is a preordered set, then a real function f on X is said to be(i) increasing if, for every x;y 2 X, [x � y ) f(x) � f(y)],(ii) order-preserving if it is increasing and, for every x;y 2 X, [x � y )f(x) < f(y)].It is well known that, if there exists a continuous order-preserving function fon a topological preordered space (X;�;�), then (X;�;�) is Nachbin separable.Finally, we recall that, given a topological space (X;�), a family G = fGr :r 2 Sg of open subsets of X is said to be a scale in (X;�) if G is a (decreasing)scale in (X;�; =).3. Existence of Continuous Order-Preserving FunctionsOur �rst aim is to characterize the existence of a real continuous order-preserving function f on a topological preordered space (X;�;�) with � IC-continuous and (X;�) normal.Theorem 3.1. Let (X;�;�) be a topological preordered space, with � IC-continuous and (X;�) normal. Then the following conditions are equivalent:(i) There exists a real continuous order-preserving function f on the space(X;�;�) with values in [0; 1];(ii) (X;�;�) is Nachbin separable;(iii) There exists a countable family fA0n;B0ngn2N of pairs of closed disjointsubsets of X such thatf(x;y) 2 X � X : x � yg � [n2N�A0n � B0n�and, for every n 2 N, if a0n 2 A0n, b0n 2 B0n, then b0n 62 d(a0n).Proof. (i) =) (ii) From considerations above, (X;�;�) is normally preordered,and therefore the implication follows from Herden [?, Corollary 4.2].(ii) =) (iii) Just observe that any countable family fA0n;B0ngn2N satisfyingthe condition of Nachbin separability also veri�es condition (iii).(iii) =) (i) Assume that condition (iii) holds, and let fA0n;B0ngn2N be acountable family of closed disjoint subsets of X with the indicated property.De�ne, for every n 2 N, An = d(A0n), Bn = i(B0n). Since � is C-continuous, Anand Bn are closed subsets of X for every n 2 N. Further, An and Bn are disjointsets for every n 2 N (otherwise, for some n 2 N there exist x 2 X, a0n 2 A0n andb0n 2 B0n such that b0n � x � a0n, and therefore b0n 2 d(a0n)). Hence, (X; �; �) is



96 G. Bosi, R. IslerNachbin separable. Since � is also I-continuous, it has been already observedthat (X; �; �) is normally preordered. Hence, from Mehta [?, Theorem 1]there exists a continuous order-preserving function f on (X; �; �) with valuesin [0; 1]. �In the sequel, a compact space is a compact-T 2 space, as in Engelking [?].Corollary 3.2. Let (X;�;�) be a topological preordered space, with � IC-continuous and (X;�) compact. Then the following conditions are equivalent:(i) There exists a real continuous order-preserving function f on the space(X;�;�) with values in [0; 1];(ii) There exists a countable family fAn;Bngn2N of pairs of compact dis-joint subsets of X such that An is decreasing, Bn is increasing, andf(x;y) 2 X � X : x � yg � [n2N(An � Bn) ;(iii) There exists a countable family fA0n;B0ngn2N of pairs of compact dis-joint subsets of X such thatf(x;y) 2 X � X : x � yg � [n2N�A0n � B0n�and, for every n 2 N, if a0n 2 A0n, b0n 2 B0n, then b0n 62 d(a0n).Proof. Observe that any compact space (X;�) is normal. Further, it is wellknown that, given a compact space, any closed subspace is compact, as well asany compact subspace is closed. Then the thesis follows from Theorem ??. �In the following theorem we provide a su�cient condition for the existenceof a continuous order-preserving function f on a topological preordered space(X;�;�), with (X;�) completely regular, and � IC-continuous.Theorem 3.3. Let (X;�;�) be a topological preordered space, with � IC-continuous and (X;�) completely regular. There exists a real continuous order-preserving function f on (X;�;�) with values in [0; 1] if the following conditionis veri�ed:(i) There exists a countable family fA0n;B0ngn2N of pairs of disjoint subsetsof X, with A0n compact and decreasing and B0n closed for every n 2 N,such that f(x;y) 2 X � X : x � yg � [n2N�A0n � B0n� :Proof. Let fA0n;B0ngn2N be a countable family of pairs of subsets of X satisfyingcondition (i) above. From C-continuity of �, i(B0n) is closed and increasing forevery n 2 N. Further, it is clear that A0n and i(B0n) are disjoint subsets of Xfor every n 2 N. Since (X;�) is completely regular, for every n 2 N thereexists a continuous function fn : X ! [0; 1] such that fn(x) = 0 on A0n andfn(x) = 1 on i(B0n) (see e.g. Engelking [?, Theorem 3.1.7]). Hence, for everyn 2 N there exists a scale G0n = fG0nr : r 2 Sng such that A0n � G0nr � X ni(B0n)for every r 2 Sn n f1g. Since � is IC-continuous, Gn = fd(G0nr) : r 2 Sng is



Separation axioms in topological preordered spaces and... 97a decreasing scale in (X;�;�) for every n 2 N (see Burgess and Fitzpatrick[?, Lemma 6.1]). De�ne, for every n 2 N, a real function fn : X ! [0; 1] byfn(x) = inffr 2 Sn : x 2 d(G0nr)g. Since it is fn(x) = inffr 2 Sn : x 2 d(G0nr)g,it is easy to show that fn is continuous. Further, fn is increasing, since for everyx;y 2 X such that x � y, fr 2 Sn : y 2 d(G0nr)g � fr 2 Sn : x 2 d(G0nr)g, andtherefore fn(x) � fn(y) from the de�nition of fn. From condition (i), for everyx;y 2 X with x � y, there exists n 2 N such that fn(x) = 0 and fn(y) = 1(see Burgess and Fitzpatrick [?, Theorem 4.1]). De�ne f = Pn2N 2�nfn. It isimmediate to observe that f is a real continuous order-preserving function on(X;�;�). �Remark 3.4. It is well known that any compact space is completely regular (seee.g. Engelking [?, Theorem 3.3.1]). So, the situation considered in Theorem ??is the most general among those considered in the paper. In the particular casewhen (X;�) is compact, condition (i) of Theorem ?? is equivalent to conditions(ii) and (iii) of Corollary ??. References[1] G. Bosi and R. Isler, Continuous order-preserving functions on a preordered completelyregular space, paper presented at II Italian-Spanish Conference on General Topology andApplications, Trieste (Italy), September 8{10, 1999.[2] D.C.J. Burgess and M. Fitzpatrick, On separation axioms for certain types of topologicalspaces, Math. Proc. Cambridge Philosophical Soc. 82 (1977), 59{65.[3] R. Engelking, General Topology (Heldermann Verlag, Berlin, 1989).[4] G. Herden, On the existence of utility functions, Mathematical Social Sciences 17 (1989),297{313.[5] G. Herden, On the existence of utility function II, Mathematical Social Sciences 18 (1989),107{111.[6] H.-P. A. K�unzi, Completely regular ordered spaces, Order 7 (1990), 283{293.[7] S. D. McCartan, Bicontinuous preordered topological spaces, Paci�c J. of Math. 38 (1971),523{529.[8] G. Mehta, Some general theorems on the existence of order-preserving functions, Mathe-matical Social Sciences 15 (1988), 135{143.[9] G. B. Mehta, Preference and utility, in: Handbook of Utility Theory, vol. 1, S. Barber�a,P. J. Hammond and C. Seidl eds. (Kluwer Academic Publishers, 1998).[10] L. Nachbin, Topology and order (D. Van Nostrand Company, 1965).Received March 2000
G. BosiDipartimento di Matematica ApplicataUniversit�a di TriestePiazzale Europa 1, 34127 TriesteItalyE-mail address: giannibo@econ.univ.trieste.it



98 G. Bosi, R. IslerR. IslerDipartimento di Matematica ApplicataUniversit�a di TriestePiazzale Europa 1, 34127 TriesteItaly