On an Extension of a Theorem of Eilenberg and a Characterization of Topological Connectedness
aa r X i v : . [ ec on . T H ] D ec On an Extension of a Theorem of Eilenbergand a Characterization of Topological Connectedness ∗ M. Ali Khan † r (cid:13) Metin Uyanık ‡ December 24, 2019
Abstract:
On taking a non-trivial and semi-transitive bi-relation constituted by two ( hard and soft ) binary relations, we report a (i) p -continuity assumption that guarantees the com-pleteness and transitivity of its soft part, and a (ii) characterization of a connected topologicalspace in terms of its attendant properties on the space. Our work generalizes antecedent re-sults in applied mathematics, all following Eilenberg (1941), and now framed in the context ofa parametrized-topological space. This re-framing is directly inspired by the continuity assump-tion in Wold (1943–44) and the mixture-space structure proposed in Herstein and Milnor (1953),and the unifying synthesis of these pioneering but neglected papers that it affords may haveindependent interest. Mathematics Subject Classification.
Key Words:
Bi-relation, non-trivial, semi-transitive, complete, transitive, connected, p -continuity,parametrically-topologized space ∗ The two theorems reported here were announced without proof at the in Berkeley, October 25, 2019; and more compre-hensively, at a departmental seminar at the
Australian National University on December 17, 2019. The authorsacknowledge with gratitude extended correspondence and conversation with Professors Max Amarante, YorgosGerasimou, Alfio Giarlotta, Farhad Husseinov, Michael Mandler, Rich McLean and Debraj Ray. Needless to say,all errors of reading and interpretation are solely the authors’. We use the certified random order in order to listthe authors; see Ray-Robson (2018, American Economic Review). † Department of Economics, Johns Hopkins University, Baltimore, MD 21218.
Email: [email protected]. ‡ School of Economics, The University of Queensland, Brisbane, QLD 4072.
Email: [email protected]: 0000-0003-0224-7851
Introduction
The question considered in this paper concerns a binary relation R on a set X conceived as asubset of X × X, with its transpose , and upper and lower section at x ∈ X respectively definedas R − = { ( x, y ) | ( y, x ) ∈ R } , R ( x ) = { y | ( x, y ) ∈ R } , and R − ( x ) = { y | ( y, x ) ∈ R } . With∆ = { ( x, x ) | x ∈ X } , R c the complement of R, its symmetric and asymmetric parts respectivelydenoted as I = R ∩ R − , and P = R \ R − , I ∩ P = emptyset, and R = I ∪ P, we let the composition R ◦ R ′ be given by ( y, x ) ∈ R ◦ R ′ if R − ( x ) ∩ R ′ ( y ) = ∅ , for any two relations R, R ′ on a X , and call a relation R on X non-trivial if P = ∅ , semi-transitive if I ◦ P ⊆ P and P ◦ I ⊆ P , transitive if R ◦ R ⊆ R , ( negatively transitive if R c is transitive), and complete if R ∪ R − = X × X . We can now ask for a sufficient condition in any register that ensures that anon-trivial, semi-transitive relation with a transitive symmetric part is complete and transitive.Thus, rather than denoting R by , as is standard especially in the social sciences, this is to askfor any register that ensures, in the vernacular of set-theory,( P = ∅ ) ∧ (( I ◦ P ⊆ P ) ∧ ( P ◦ I ⊆ P )) ∧ ( I ◦ I ⊆ I ) = ⇒ ( R ∪ R − = X × X ) ∧ ( R ◦ R ⊆ R ) . In his remarkable paper, Eilenberg (1941) provided a partial answer this question by invok-ing a topological register and assuming a continuous binary (preference) relation on a connected(choice) set. His answer inaugurated the study of a partially ordered topological space, and hiswork received important substantive extension at the hands of Debreu (1954, 1964), Ward (1954aand 1954b), Sonnenschein (1965, 1967), McCartan (1966) and Schmeidler (1971). In the twin-registers of economic and decision theory, the extension involved a move to a setting of totalpre-orders, which is to say from a setting of singleton indifference sets to a more general situationwhere the preference map of a consumer is delineated by indifference surfaces.Recent work in both mathematical economics and mathematical psychology has rediscov-ered these original papers, and under the label of the Eilenberg-Sonnenschein (ES) research pro-gram, has been particularly stimulated by what it sees as the derivation of behavioral conse-quences of merely technical topological assumptions; see, for example, Khan and Uyanık (2019),Giarlotta and Watson (2019) and their references. Eilenberg, and following him Ward, phrasedtheir results only in the language of topological structures, but Herstein and Milnor (1953), withWold (1943–44) as their important precursor, focused on the functional representation of therelation, and shifted all topological assumptions on the choice set to that on the unit inter-val. This is to say, they focused their attention on the mixing operation, rather than on theobjects of choice itself. Wold (1943–44) used a similar scalar-continuity property in his work,but with an additional monotonicity assumption; also see Fishburn (1982), Wakker (1989),Bridges and Mehta (1995), Herden and Pallack (2001) and Candeal, Indur´ain and Molina (2012),Galaabaatar, Khan and Uyanık (2019) on numerical representation of preferences. In addition tothis parametrized topological setting, we also mention for the record, a third setting, that involvesno topology at all, but constrains itself to a purely algebraic structure. This goes back at least This notation is perhaps original to (Arrow, 1951, p. 11) who observed that he was representing “preference[relations] by a notation not customarily employed in economics, though familiar in mathematics and particularlyin symbolic logic.”
1o seminal paper of Holder (1901); also see Fishburn (1972) and Luce (2000).In this paper we work with a richer order-theoretic structure defined by two binary rela-tions instead of one, a structure that is only now being appreciated and given prominence; seeGiarlotta and Greco (2013), Giarlotta (2014), Giarlotta and Watson (2019) and Uyanık r (cid:13)
Khan(2019a) and their references; also see Chipman (1971) and Maccheroni, Marinacci and Rustichini(2006) for an implicitly assumed bi-preference structure. In this richer structure, we introduce aparametric continuity concept for (bi-)relations that does not assume any structure on the choiceset itself, and is weaker than the usual continuity properties that require it to have one of thetopological or algebraic structures. We report two theorems, one of which provides a new charac-terization of topological connectedness, and show by examples that our results are non-vacuous.Even in the special case of a single binary relation, our first result provide a synthetic treatmentof the antecedent ES literature by generalizing and unifying it, and in particular, our analyticaltreatment provides an alternative proof of the theorems of Eilenberg (1941), Sonnenschein (1965,1967) and Rader (1963).
We elaborate the terminology presented in Section 1 to the setting of a bi-relation.
Definition 1.
A bi-relation on a set X is a pair ( R H , R S ) of relations on X such that R H ⊆ R S and P S ⊆ P H . A bi-relation ( R H , R S ) is non-trivial if P S = ∅ , and semi-transitive if (i) R H is semi-transitive, (ii) I S ◦ R H ⊆ R S and R H ◦ I S ⊆ R S , (iii) P H ∩ ( P S ◦ I S ) ⊆ P S and P H ∩ ( I S ◦ P S ) ⊆ P S . For any bi-relation ( R H , R S ), we informally refer to R H as its hard part, and R S as its soft part.We now present the basic continuity assumption that motivates this paper and is the subject ofits investigation. Definition 2.
A bi-relation ( R H , R S ) on a set X is p -continuous (parametrically continuous)if for all x, y ∈ X , there exist a topological space Λ xy and a function f xy : Λ xy → X with x, y ∈ f xy (Λ xy ) such that for all z ∈ X , (i) f − xy ( R H ( z )) and f − xy ( R − H ( z )) are closed, and (ii) f − xy ( P S ( z )) and f − xy ( P − S ( z )) are open. We call a p -continuous bi-relation connected if each space Λ xy in Definition 2 is connected.The concept is admittedly abstract, but one can get a feel for it by dissociating the two aspectsof Herstein and Milnor (1953) that it generalizes: first, keeping the unit interval, and focusing onthe fact that rather than a linear function, any function is being considered; and second, replacingthe unit interval by any topological space, both operations localized by the dependence on thepair (Λ xy , f xy ) on the two chosen points, x and y .We now recall for the reader the standard2 efinition 3. A topological space X is connected if it is not the union of two non-empty, disjointopen sets. A subset of X is connected if it is connected as a subspace. A relation R on a set X is p -continuous if the bi-relation ( R, R ) is p -continuous, and it is connected if the bi-relation ( R, R ) is connected. The reader may wish to contrast this definition to Fishburn(1972, p. 27). Moreover,
Definition 4.
A relation on a topological space is continuous if it has closed sections and itsasymmetric part has open sections.
We can now present our first result,
Theorem 1.
The soft part of every non-trivial, semi-transitive and connected bi-relation is com-plete and transitive.
Next, we present a special case of Theorem 1 for a binary relation. Note that a relation R on a set X is non-trivial, semi-transitive, connected and has a transitive symmetric part if andonly if the bi-relation ( R, R ) on X is non-trivial, semi-transitivity and connected. On setting R H = R S in a bi-relation ( R H , R S ) , we obtain the following as a corollary of Theorem 1. Corollary 1.
Every non-trivial, semi-transitive and connected binary relation whose symmetricpart is transitive, is complete and transitive.
In the light of the above discussion, we can now present
Theorem (Eilenberg 1941).
Every anti-symmetric, complete and continuous relation on aconnected set is transitive.
Theorem 1 provides a generalization and alternative proof not only of Eilenberg (1941, 2.1),but also of the extensions pursued in Rader (1963) and Sonnenschein (1965). Without anyattempt to downplay the importance of the exercise, we invite the reader to show that theprincipal results in the literature follow as a consequence of Theorem 1 and Corollary 1 above,and Lemma 1 and Propositions 1–2 below: Rader (1963, Lemma), Sonnenschein (1965, Theo-rem 3), Schmeidler (1971, Theorem), Dubra (2011, Theorem 1), Karni and Safra (2015, The-orem 1), McCarthy and Mikkola (2018, Theorem 1), Khan and Uyanık (2019, Proposition 1);Giarlotta and Watson (2019, Theorem 5.2) and finally Uyanık r (cid:13)
Khan (2019b, Theorems 1 and2). In our judgment, this exercise testifies to the importance of the results presented here. (Note,however, that the authors of the above papers present their results in an extended form that alsoinvolve consideration not germane to those pursued here; our generalization concerns only therelevant part of their results.) In the sequel, we also illustrate the novelty of our proof-techniquesby contrasting them with earlier proofs.We now turn to the proofs, and begin by recalling the following.
Lemma 1.
For any binary relation R on a set X the following is true: if R is complete and semi-transitive, then I is transitive; (b) if P is negatively transitive, then it is transitive and R is semi-transitive; (c) R is transitive if and only if it is semi-transitive and P, I are transitive.
For a proof, see Sen (1969, Theorem I) and Khan and Uyanık (2019, Proposition 2) on Sen’sdeconstruction of the transitivity postulate.We shall also need the following two claims concerning a non-trivial, semi-transitive andconnected bi-relation ( R H , R S ) on an arbitrary set X . Claim . If ( y, x ) ∈ P S , then P S ( y ) ∪ P − S ( x ) = X .Claim . R S ∪ R − S = X × X . The proof of Theorem 1 follows as a direct consequence of these preliminary results
Proof of Theorem 1.
Claim 2 already establishes the completeness of the soft part of the bi-relation. Now negative transitivity of P S is equivalent to Claim 1. Then (b) of Lemma 1 implies R S is semi-transitive and P S is transitive. It follows from Claim 2, semi-transitivity of R S and(a) of Lemma 1 that I S is transitive. Then (c) of Lemma 1 implies R S is transitive.All that remains now are the proofs of the two claims. Proof of Claim 1.
Pick ( y, x ) ∈ P S . Then ( y, x ) ∈ P H . It follows from semi-transitivity of thebi-relation that R H ( y ) ∪ R − H ( x ) = P S ( y ) ∪ P − S ( x ) . (1)One direction of the inclusion relationship immediately follows from the definition of bi-relation.In order to prove the other direction pick z ∈ R − H ( x ). Then either z ∈ I H ( x ) or z ∈ P − H ( x ).Assume z ∈ I H ( x ). It follows from the definition of bi-relation that z ∈ I S ( x ). Moreover, x ∈ P H ( y ), z ∈ I H ( x ) and semi-transitivity of R H imply ( y, z ) ∈ P H . Note that z ∈ I S ( x ) and x ∈ P S ( y ) imply ( y, z ) ∈ I S ◦ P S . Then semi-transitivity of the bi-relation implies z ∈ P S ( y ). Nowassume z ∈ P − H ( x ). Then the definition of bi-relation implies that either x ∈ P S ( z ) or x ∈ I S ( z ).If x ∈ P S ( z ), then there is nothing to prove. Now assume x ∈ I S ( z ). Then semi-transitivityof the bi-relation, z ∈ I S ( x ) and x ∈ P H ( y ) imply that z ∈ R S ( y ). Hence either z ∈ P S ( y )or z ∈ I S ( y ). If z ∈ I S ( y ), then semi-transitivity of the bi-relation, x ∈ P S ( y ) , y ∈ I S ( z ) and x ∈ P H ( z ) imply that x ∈ P S ( z ). This contradicts x ∈ I S ( z ). Hence, z ∈ P S ( y ) must hold. Theproof for z ∈ R H ( y ) is analogous.We next prove that P S ( y ) ∪ P − S ( x ) = X . To this end pick z ∈ X . It follows from theconnectedness of the bi-relation that there exist a connected topological space Λ yz and a function f yz : Λ yz → X satisfying the conditions in Definition 2. It follows from Equation 1 above that f − yz ( R H ( y )) ∪ f − yz ( R − H ( x )) = f − yz P S ( y ) ∪ f − yz ( P − S ( x )) . (2)It follows from x ∈ P S ( y ) and y = f yz ( λ ) for some λ ∈ Λ yz that λ ∈ f − yz ( P − S ( x )). Hencethe set in Equation 2 is non-empty. It follows from p -continuity that it is both closed andopen in Λ yz . Therefore, as a nonempty, closed and open subset of a connected set Λ yz , the set f − yz ( P S ( y )) ∪ f − yz ( P − S ( x )) is equal to Λ yz . Then it follows from z = f yz ( δ ) for some δ ∈ Λ yz that δ ∈ f − yz ( P S ( y )) ∪ f − yz ( P − S ( x )), hence z ∈ P S ( y ) or z ∈ P − S ( x ).4 roof of Claim 2. Assume there exists u, v ∈ X such that ( u, v ) / ∈ R S ∪ R − S . Note that non-triviality of R S implies that (¯ y, ¯ x ) ∈ P S for some ¯ x, ¯ y ∈ X . Since P S is negatively transitive, then u ∈ P − S (¯ x ) or u ∈ P S (¯ y ). Assume u ∈ P − S (¯ x ). Then negative transitivity of P S implies that v ∈ P − S (¯ x ) or v ∈ P S ( u ). Since u, v / ∈ R S ∪ R − S , therefore v ∈ P − S (¯ x ). Hence ¯ x ∈ P S ( u ) ∩ P S ( v ).The semi-transitivity of the bi-relation implies that I H ( u ) ∩ R H ( v ) = ∅ = R H ( u ) ∩ I H ( v ) . (3)In order to prove the first equality, assume there exists z ∈ I H ( u ) ∩ R H ( v ). If z ∈ P H ( v ), thensemi-transitivity of R H implies that u ∈ P H ( v ), hence u ∈ R S ( v ). This yields a contradiction.Then assume z ∈ I H ( v ). The definition of bi-relation implies that z ∈ I S ( v ). Then it followsfrom semi-transitivity of bi-relation that ( u, v ) ∈ R S . This yields a contradiction. The proof ofthe second equality is analogous.Therefore, Equation 3 and the definition of the bi-relation imply that R H ( u ) ∩ R H ( v ) = P S ( u ) ∩ P S ( v ) . (4)It follows from the connectedness of the bi-relation that there exist a connected topological spaceΛ ¯ xu and a function f ¯ xu : Λ ¯ xu → X satisfying the conditions in Definition 2. It follows fromEquation 4 above that f − xu ( R H ( u )) ∩ f − xu ( R H ( v )) = f − xu ( P S ( u )) ∩ f − xu ( P S ( v )) . (5)It follows from ¯ x ∈ P S ( u ) ∩ P S ( v ) and ¯ x = f ¯ xu ( λ ) for some λ ∈ Λ ¯ xu that λ ∈ f − xu ( P S ( u )) ∩ f − xu ( P S ( v )). Analogously it follows from u / ∈ P S ( u ) and u = f ¯ xu ( δ ) for some δ ∈ Λ ¯ xu that δ / ∈ f − xu ( P S ( u )) ∩ f − xu ( P S ( v )). Hence the set if Equation 2 is non-empty and proper subset ofΛ yz . It follows from p -continuity that it is both closed and open in Λ yz . This contradicts withthe connectedness of Λ ¯ xu .The proof is analogous for u ∈ P S (¯ y ). Therefore R S is complete.We now turn to the second main result of this paper. Note that the essential point ofTheorem 1 is that its conclusion of completeness and transitivity of the soft relation do notcall for any mathematical structure on the (choice) set on which the (preference) bi-relation isdefined: a local topological structure on the local parameter space suffices enough to obtain.We now work towards a converse question first posed in the context of a single binary relationin Khan and Uyanık (2019). Our second theorem shows that topological connectedness is bothnecessary and sufficient for completeness and transitivity of a bi-relation. Towards this end, forany topology τ on X , let R ( X, τ ) , denote the set of all non-trivial and semi-transitive bi-relations( R H , R S ) on X such that τ ( R H , R S ) ⊆ τ , where τ ( R H , R S ) denotes the coarsest topology on X containing all sections of P S and the complements of the sections of R H . We can then present Theorem 2.
For any set X and topology τ on it the following is equivalent: (a) the space ( X, τ ) is connected; (b) if ( R, R ) ∈ R ( X, τ ) , then R is complete and transitive; (c) if ( R H , R S ) ∈ R ( X, τ ) , then R S is complete and transitive.Proof. The implication (a) ⇒ (c) follows from Theorem 1 above and Proposition 1 below,(c) ⇒ (b) from setting R H = R S and (b) ⇒ (a) from Khan and Uyanık (2019, Theorem 2).5e note that in a parallel but different path, McCartan (1966) provides a characterizationof compactness and Hausdorff separation axiom by using the preferences defined on a choice set.We have already observed above that the richer bi-preference structure that we work withenables different techniques of proof. We elaborate this observation here, beginning with Eilen-berg’s method . As a preliminary remark, note that although Eilenberg assumes the relation to becomplete, what follows shows that his method-of-proof extends to a situation where this is notso. Given his assumption on R being an anti-symmetric and continuous relation on a connectedset X, R ∩ R − ⊆ ∆. The method requires one to pick x, y, z ∈ X such that ( z, y ) , ( y, x ) ∈ P .The connectedness of X and the continuity of R imply that R ( y ) is connected and contains y . Note first that if y / ∈ R ( y ), then x ∈ R ( y ) , z / ∈ R ( y ) and continuity of R imply that R ( y )is a closed, open, non-empty and a proper subset of X , which contradicts the connectednessof the space. Now assume R ( y ) is disconnected. Then there exists non-empty, disjoint andclosed subsets A, B of the subspace R ( y ) such that A ∪ B = R ( y ). Let y ∈ B . Note that X = A ∪ ( B ∪ R − ( y ) ∪ ( R ( y ) ∪ R − ( y )) c ∪ { y } ) = A ∪ ( B ∪ R − ( y ) ∪ ( P ( y ) ∪ P − ( y )) c ), where A and the set in paranthesis are non-empty, disjoint and closed. This contradicts the connectednessof X . Finally, note that R ( y ) = P ( z ) ∪ P − ( z ) ∪ ( R ∪ R − ) c ( z ) , and recall that x, y ∈ R ( y )and y ∈ P ( z ). If x / ∈ P ( z ), then this furnishes us a contradiction to the connectedness of R ( y ).Therefore, ( z, x ) ∈ P . Since R is anti-symmetric, it is transitive. The proof is complete.Now consider the special case of the method we pursue in Theorem 1 which begins with ananti-symmetric and continuous relation R on a connected set X , and picks x, y ∈ X such that( y, x ) ∈ P . Then the anti-symmetry of R implies that R ( y ) ∪ R − ( x ) = P ( y ) ∪ P − ( x ). Thereforeit follows from the continuity of R and connectedness of X that P ( y ) ∪ P − ( x ) = X . Hence P is negatively transitive, which implies P is transitive. In order to see this, pick ( y, z ) , ( z, x ) ∈ P .Negative transitivity of P implies that ( y, x ) ∈ P or ( z, y ) ∈ P . Therefore ( y, x ) ∈ P , hence P istransitive. Since R is anti-symmetric, it is transitive. Therefore the proof is complete.Finally, we leave it to the reader to check that our method of proof has some similaritieswith the argument pursued by Schmeidler (1971), but there are also some subtle differences. Inany case, is not the intricacy of the proofs but the surprise of the conclusion that is the point ofall this work.We conclude this section by listing possible future directions of research. Note that p -continuity does not require f xy to be mixture-linear, and more importantly, for all x, y the functionis allowed to be chosen differently. This has relevance to Diewert, Avriel and Zang (1981) whichstudies different forms of convexity assumptions on functions, and merits investigation. It willalso be interesting to extend the results in this paper to k -connected spaces and component-wisenon-trivial relations on an arbitrary topological spaces; see for example Khan and Uyanık (2019).Furthermore, given our tilt to the subject, generalizing Lorimer’s (1967) set-theoretic version ofEilenberg and Sonnenschein’s result is of interest. Finally, it will be interesting to explore howour results generalize to n -ary relations, given their importance in the literature of analyticalphilosophy; see Anand (1993); Temkin (1996, 2015) and their references.6 Some Additional Considerations
In this section, we turn to the usual scalar and section continuity properties of a relation, and showthat they are stronger than p -continuity, once relevant mathematical structures on the choice setare in place; see Uyanık r (cid:13) Khan (2019a) for a detailed discussion of the relationship betweendifferent continuity assumptions on preferences, and Ciesielski and Miller (2016) for those onfunctions.
Proposition 1.
Every continuous relation on a topological space is p -continuous.Proof. Let R be a continuous relation on a topological space X . For all x, y ∈ X , setting Λ xy = X and f xy ( a ) = a for all a ∈ Λ xy finishes the proof. Finally, note that if X is connected, then thebi-relation is connected.For scalar continuity concepts, we say that a set S is a mixture set if for any x, y ∈ S and forany µ ∈ [0 ,
1] we can associate another element, xµy, that is in S , and where for all λ, µ ∈ [0 , x y = x, (S1) xµy = y (1 − µ ) x, and (S3)( xµy ) λy = x ( λµ ) y. Definition 5.
We refer a relation R on a mixture set S to be (i) mixture-continuous if for all x, y, z ∈ S , the sets { λ ∈ [0 , | xλy ∈ R ( z ) } and { λ ∈ [0 , | xλy ∈ R − ( z ) } are closed, (ii) Archimedean if for all x, y, z ∈ S with ( y, x ) ∈ P , there exists λ, δ ∈ (0 , such that xλz ∈ P ( y ) and yδz ∈ P − ( x ) .Moreover, R is scalarly continuous if it is mixture-continuous and Archimedean. Proposition 2.
Every scalarly continuous relation on a mixture set is connected, hence p -continuous.Proof. Let R be a scalarly continuous relation on a mixture set X and Λ xy = [0 ,
1] for all x, y ∈ X .Then, for all x, y ∈ X , f xy ( λ ) = xλy for all λ ∈ Λ xy . It is easy to observe that the closedness of thesets f − xy ( R ( z )) and f − xy ( R − ( z )) is equivalent to mixture-continuity property. Under the mixturecontinuity assumption, it follows from Galaabaatar, Khan and Uyanık (2019, Proposition 1) thatArchimedean property is equivalent to the property that f − xy ( P ( z )) and f − xy ( P − ( z )) are open.The following two definitions extends the usual continuity assumptions on uni-relations tobi-relations. Definition 6.
A bi-relation ( R H , R S ) on a topological space is continuous if R H has closed sectionsand P S has open sections. Definition 7.
A bi-relation ( R H , R S ) on a mixture set S is scalarly continuous if R H is mixturecontinuous and for all x, y, z ∈ S , the sets { λ ∈ [0 , | xλy ∈ P S ( z ) } and { λ ∈ [0 , | xλy ∈ P − S ( z ) } are open. The second part of scalar continuity is a property slightly stronger than R S being Archimedean.The Archimedean property and part (ii) of Definition 7 are equivalent if R S is mixture-continuous;see Galaabaatar, Khan and Uyanık (2019, Proposition 1) for details. We leave it to the reader toobtain the versions of the two propositions above for bi-relations.7 Some Concluding Examples
We conclude the paper with three examples. The first is a modified version of a famous exampledue to Genocchi and Peano (1884), and illustrates a preference relation that is p -continuous butnot continuous, thus ensuring that the two theorems reported in this paper are non-vacuous. Thesecond example concerns the canonical order in R n ; it satisfies all of the assumptions of Theorem1 except the openness requirements of the sections. The third example is a simple modificationof the second that is phrased in terms of a relation that is both incomplete and non-transitive. Example 1.
Let X = [0 , and f : X → R defined as f ( x ) = 2 x x x + x if x = (0 ,
0) and f (0 ,
0) = (1 , . Note that f is not continuous in each variable, and hence not (jointly) continuous. Induce arelation R on X such that ( x, y ) ∈ R if f ( x ) ≤ f ( y ). It is clear that R is complete and transitive.However, R is not mixture-continuous, hence not continuous. In order to see this, let x = (0 , y = (1 , { λ ∈ [0 , | λx + (1 − λ ) y ∈ R − ( y ) } = (0 ,
1] is not open in X . It is notdifficult to show that R is p -continuous; simply for x, y above, set Λ xy = [0 ,
1] and f xy ( λ ) = ( λ, λ )for λ < . f xy ( λ ) = ( λ, − λ ) for λ ≥ . Example 2.
Let X = R n + and R be the usual relation ≥ on X defined as “ x ≥ y if x i ≥ y i forall i .” The asymmetric part P of R is the relation > defined as “ x > y if x i ≥ y i for all i and x j > y j for some j .” It is clear that X is connected, R is anti-symmetric, transitive, non-trivialand has closed sections. However the section of P are not open since P ( x ) = R ( x ) \{ x } and P − ( x ) = R − ( x ) \{ x } . Clearly, R is incomplete. Example 3.
Let X = [0 ,
1] and R = { ( x, y ) | x ≤ . , x ≤ y ≤ . } ∪ { ( x, y ) | x ≥ . , y ≥ x } .It is clear that R is reflexive and anti-symmetric, hence R is semi-transitive and has a transitiveindifference. Note that R − (0 .
5) = [0 , .
5] and R (0 .
5) = [0 . , , . × [0 . , * R implies R is not transitive. It is clear that R is not complete. Since R has closed graph, it hasclosed sections. However, P does not have open sections. For example P (0 .
25) = (0 . , .
5] and P − (0 .
75) = [0 . , .
75) which are not open in [0,1].
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