On Continuity of Incomplete Preferences∗† Georgios Gerasimou‡ April 25, 2012

Abstract A weak (strict) preference relation is continuous if it has a closed (open) graph; it is hemicontinuous if its upper and lower contour sets are closed (open). If preferences are complete these four conditions are equivalent. Without completeness continuity in each case is stronger than hemicontinuity. This paper provides general characterizations of continuity in terms of hemicontinuity for weak preferences that are modeled as (possibly incomplete) preorders and for strict preferences that are modeled as strict partial orders. Some behavioral implications associated with the two approaches are also discussed.

Keywords: Incomplete Preferences; Continuity; Hemi-Continuity; Preorders; Strict Partial Orders.

JEL Classification: D01

∗ Dedicated

to Konstantinos Gerasimou. paper is based on Chapter 1 of my PhD thesis at the University of Cambridge. I am grateful to Robert Evans for his careful readings of previous versions and to a referee of this journal for comments and questions that led to several improvements. Thanks also to three more referees and to John Quah for their thoughtful remarks. I am solely responsible for any errors. ‡ School of Economics & Finance, University of St Andrews. Email address: [email protected] † This

1

Introduction

Incomplete preference orderings are useful in decision theory when the aim is to model an agent’s indecisiveness, as expressed by the existence of preference-incomparable alternatives. They also arise naturally in social choice theory when the goal is to capture the common component of the generally conflicting preferences of a group of individuals, for example by means of the Pareto relation.1 In both cases, two approaches can be followed to model incompleteness. The first approach takes weak preferences as primitive, while the second approach builds on strict preferences instead. A theoretical distinction between (individual or social) indifference and incomparability can be achieved with the first approach, but not with the second. Although theories of the former type have a richer structure and may, therefore, appear to be unambiguously better than those of the latter, the analysis in the present paper leads to a more nuanced conclusion, at least insofar as the preference domain requires the use of continuity axioms on preferences. More specifically, this paper studies the continuity axioms that are associated with the two approaches and, as a corollary to a theorem of Schmeidler (1971), it first shows that the weak-preference approach leads to counter-intuitive behavioral predictions. In particular, irrespective of whether the weak preference relation is hemicontinuous (i.e. with closed upper and lower sections) or continuous (i.e. with a closed graph), as long as its domain is a connected space and the relation is nontrivial there are alternatives that are ranked by strict preference and yet are also such that arbitrarily neighboring alternatives are incomparable. By contrast, this prediction, which seems to be neither descriptively accurate nor normatively sound, is avoided when the strict-preference approach is followed. In this case, strict-preference continuity (i.e. the open-graph property) ensures that such discontinuities do not occur. The main contribution of this paper is the characterization of continuity in terms of hemicontinuity for incomplete weak and strict preferences. When the weak preference relation is complete and the strict preference relation is its asymmetric part, all four notions of (hemi)continuity are equivalent. When completeness is removed and weak preferences are merely reflexive and transitive, however, continuity in both cases is stronger than hemicontinuity, while the two groups of axioms are logically distinct. Since in both situations hemicontinuity is the weaker axiom but continuity is the one that tends to be useful in practice,2 it is naturally of interest to know what are the additional properties that are implied by continuity in each case and which, together with hemicontinuity, also imply the stronger axiom. I identify these conditions, which, as it turns out, place restrictions on the behavior of the preference relation’s complement, i.e. on the relations “not weakly preferred to” and “not preferred to”, respectively. While the step from hemi- to full continuity is “easy” for the strict-preference approach in the sense that it is achievable by means of intuitive behavioral assumptions, this does not seem to be the case for the approach that is based on weak preferences. More specifically, in light of the newly identified condition that is involved in the characterization of weak-preference continuity, in order to obtain this axiom from its weaker 1 For a recent investigation of the existence of social welfare functions when social preferences are incomplete in general, see Stecher (2008). 2 In decision theory, Ghirardato et al (2003) and Dubra et al (2004) are examples where weak-preference continuity is used, while Bewley (1986/2002), Manzini and Mariotti (2008) and Gerasimou (2010) are ones where strict-preference continuity is used instead (in the case of Bewley’s paper, continuity is implied by the assumptions of hemicontinuity and independence; see section 4 below).

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counterpart it suffices to assume that the weak-preference relation’s complement is distinctly transitive. While this condition is arguably behavioral, it generates unreasonable predictions (for instance, it implies distinct transitivity of the incomparability relation). Although the condition is not necessary for the above purpose, it is not clear whether it can be replaced by more plausible behavioral assumptions, as in the case of strictpreference continuity.

2

Notation and Preliminaries

Let X be a set of alternatives and % ⊆ X × X a binary relation on X. When (x, y) ∈ % is the case, I write x % y. The relation % captures weak preferences and is assumed reflexive (x ∈ X implies x % x) and transitive (x, y, z ∈ X and x % y, y % z implies x % z), hence a preorder. Let  and ∼ be the asymmetric and symmetric components of %, capturing strict preferences and indifference, respectively. The former is asymmetric (x  y implies y 6 x) and transitive (hence a strict partial order ) while the latter is reflexive, symmetric (x ∼ y implies y ∼ x) and transitive (hence an equivalence relation). The complement of % is 6% := (X × X) \ %, while the incomparability relation associated with % is f := {(x, y) : x 6% y, y 6% x}. It is clearly true that 6% ≡ ≺ ∪ f, where ≺ is the inverse of , and f ≡ 6% ∩ 6-, where 6- is the inverse of 6%. By definition, 6% is irreflexive and f is irreflexive and symmetric. The relation % is complete if for all x, y ∈ X, x % y or y % x holds. Obviously, % is complete if and only if f ≡ ∅, which in turn is true if and only if 6% ≡ ≺. The relation % is said to be non-trivial if  6= ∅ and order-dense if x  y implies x  z  y for some z ∈ X. The upper and lower contour sets (or sections) of % relative to an element x ∈ X are the sets U% (x) := {y ∈ X : y % x} and L% (x) := {y ∈ X : x % y}, while the indifference and incomparability classes of x are I(x) := {y ∈ X : y ∼ x} and J(x) := {y ∈ X : x f y}, respectively. The upper and lower sections of a strict partial order  are defined analogously. Notice that a preorder induces a strict partial order in a natural way by means of its asymmetric component, whereas a strict partial order does not automatically extend to a preorder. When I have in mind the specific strict partial order induced by a preorder % I will always use the notation . In contrast, an abstract strict partial order will be denoted by >, unless otherwise noted. If X is a topological space, then the complement, closure, interior and boundary of a set A ⊂ X are denoted by Ac , cl(A), int(A) and bd(A), respectively. Furthermore, an open neighborhood of a point x ∈ X will be denoted by N (x), while N (x, y) will stand for an open neighborhood of the pair (x, y) ∈ X × X. Some reference is made below to T1 , T2 , connected, compact and first-countable topological spaces. The reader is referred to Aliprantis and Border (2006) for definitions and analysis of the properties of these spaces. The commonly used preference (hemi)continuity properties that are this paper’s main object of study can now be formally stated. Definition 1 A strict partial order > on a topological space X is: 1. (Upper- and Lower-)Hemicontinuous, if U> (x) and L> (x) are open for all x ∈ X; 2. Continuous, if > is an open subset of the product space X × X. 2

Definition 2 A preorder % on a topological space X is: 1. (Upper- and Lower-)Hemicontinuous, if U% (x) and L% (x) are closed for all x ∈ X; 2. Bi-hemicontinuous, if both % and  are hemicontinuous; 3. Continuous, if % is a closed subset of the product space X × X. As already mentioned, a continuous preorder is hemicontinuous, but without completeness the converse need not be true. Continuity, however, does not imply bihemicontinuity. Furthermore, a continuous strict partial order is hemicontinuous but the converse is not true in general. Yet, the following well-known result shows that the implication holds both ways if the strict partial order is the asymmetric part of a complete preorder, and also establishes that all the above (hemi)continuity conditions become equivalent in this case. Theorem 1 The following are equivalent for a complete preorder % on a topological space X. 1.  is continuous; 2.  is hemicontinuous; 3. % is continuous; 4. % is hemicontinuous. The reader is referred to Ward (1954, Lemma 3), Bergstrom et al (1976, Theorem 2) and Bridges and Mehta (1995, Proposition 1.6.2) for three proofs of this result.

3 3.1

(Hemi)Continuous Preorders Fragility

I start the analysis of weak-preference (hemi)continuity by first showing that, under minimal regularity conditions on the domain and on the weak preference relation, hemicontinuity (and therefore continuity too) generates unreasonable behavioral predictions. This conclusion comes about as an implication of the following remarkable result due to Schmeidler (1971).3 Theorem 2 (Schmeidler) A nontrivial bi-hemicontinuous preorder on a connected topological space is complete. To see the relevance of the domain condition in Schmeidler’s theorem, recall that most of the traditionally important choice domains in economics (e.g. the consumption set Rn+ in neoclassical consumer theory, a probability simplex in choice under risk, a convex subset of a space of functions in choice under uncertainty) are convex subsets of some real vector space, and therefore also connected when endowed with their usual inherited topologies. In light of Schmeidler’s theorem, and in order to formalize the claim made above, I introduce the following definition: 3 Dubra (2011) proved a result similar to Schmeidler’s, showing that, when defined in a domain of lotteries, a preorder that satisfies independence, Archimedean continuity and mixture continuity also satisfies completeness.

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Definition 3 A preorder % on a topological space X is fragile if there exist x, y ∈ X with the following properties: 1. x  y; 2. Every neighborhood N (x, y) ⊂ X × X contains a pair (x0 , y 0 ) such that x0 f y 0 . In words, the weak preference relation is fragile if the individual is able to rank two alternatives x and y by strict preference and yet arbitrarily small perturbations of one (or both) of these alternatives make her indecisive. Put differently, it is fragile if the strict preference relation it induces is not always robust against indecision. The following result is a corollary to Schmeidler’s theorem and shows that hemicontinuity and nontriviality are sufficient conditions for an incomplete preorder to be fragile in many domains of interest. Corollary 3 A hemicontinuous, non-trivial, incomplete preorder on a connected topological space is fragile. Proof: 4 In view of Schmeidler’s theorem, the stated assumptions imply that there exists x ∈ X such that L (x) or U (x) is not open. Without loss of generality, suppose that L (x) is not open. Then there exist y ∈ L (x) and a net (yα )α∈D , such that yα → y and x  yα for α large enough. Suppose yα ∈ U% (x) for α large enough. Since U% (x) is closed and yα → y, this implies y % x, a contradiction. Thus, for α large enough it holds that yα ∈ U% (x)c . Together with yα ∈ L (x)c , this implies yα ∈ L% (x)c for α large enough. Thus, for α large enough, it holds that yα ∈ L% (x)c ∩ U% (x)c ≡ J(x). Since J(x) is open, there is a neighborhood of yα contained in J(x). Finally, since for every neighborhood N (yα , x) in X × X there exist neighborhoods N (yα ), N (x) in X such that N (yα ) × N (x) ⊂ N (yα , x) and it is also true that N (yα ) ∩ J(x) 6= ∅ for every such N (yα ), fragility of % is established.  Example 1 Let ≥ be the usual partial ordering5 in R2 , i.e. x ≥ y if xi ≥ y i for all i = 1, 2. This relation is a hemicontinuous (and also continuous) preorder that is nontrivial, while its domain is a connected space. To show that it is fragile, let > be the asymmetric part of ≥, i.e. x > y if x ≥ y and xi > y i for some i ∈ {1, 2}. Clearly, x f y if xi > y i and xj < y j for i 6= j. Let x = (a, b) and y = (a, c), with b > c. It holds that x > y. Let B (x) be the Euclidean open ball about x with radius  > 0. For all δ ∈ (0, ) the vector x0 = (a − δ, b) is in B (x) and also such that x0 f y. ♦ From a normative point of view, fragility of a preference relation is an undesirable property, both when this preference relation is that of an individual and also when it represents the preferences over social alternatives that a group of individuals have in common (in the latter case these preferences are obviously defined on suitably restricted domains of social alternatives). Indeed, one would expect that when decision makers 4I

thank a referee of this journal for suggesting this shorter proof. partial order ≥ is a preorder that is also antisymmetric, i.e. x ≥ y and y ≥ x implies x = y.

5A

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express strict preference for one alternative over another, marginal changes in these two alternatives should not result in them becoming incomparable. If they do, then doubt should perhaps be cast on the validity of the strict-preference comparison between the original alternatives. Finally, introspection and casual empiricism do not seem favorable for the property’s descriptive accuracy either.

3.2

Characterization

The main contribution of the paper is the characterization of continuity in terms of hemicontinuity for a preorder defined on an arbitrary space, which is the task carried out in the present subsection. Recall first that, as also noted by Ward (1954, p. 148), an order topology on a space X that is induced by a preorder % on X can be determined by assuming mere hemicontinuity of %, by taking as a subbase for closed sets in X the collection {A ⊂ X : A = U% (x) or A = L% (x) for some x ∈ X}. Furthermore, the topology generated by this subbase is the coarsest on X for which % is hemicontinuous. Clearly then, in the case where % is incomplete and hemicontinuity does not imply continuity, the latter axiom imposes more structure than an order topology on X.6 This additional structure is pinned down by the characterization offered below. In particular, it turns out that studying the behavior of the relation 6% can provide key insights on the interplay between continuity and hemicontinuity of the preorder % in the general case where % is not complete. In this connection, the following novel concept is introduced next, which plays a central role in what follows. Definition 4 The relation 6% expands local transitivity if for all x, y, z ∈ X such that x 6% z, z % 6 y and x 6% y there exist open neighborhoods N (x) and N (y) such that x0 6% z, z 6% y 0 and x0 6% y 0 for all x0 ∈ N (x) and y 0 ∈ N (y). The following theorem shows that a preorder on an arbitrary topological space is continuous if and only if it is hemicontinuous and its complement expands local transitivity. The approach followed in the second part of the proof was originally inspired by an argument that appears in the proof of Theorem 2 in Bergstrom et al (1976).7 Theorem 4 The following are equivalent for a preorder % on a topological space X: 1. % is continuous; 2. % is hemicontinuous and 6% expands local transitivity. Proof: 6 It is precisely in this sense that hemicontinuity is considered to be a more primitive axiom than continuity in this paper. 7 Theorem 2 in Bergstrom et al (1976) states that if the complement of a strict partial order is transitive (i.e. a preorder), then the strict partial order is hemicontinuous if and only if it is continuous. However, a strict partial order’s complement can be a preorder only if it is in fact a complete preorder (Proof: Let > be a strict partial order and ≯ a preorder. If ≯ is incomplete, then ¬(x ≯ y) and ¬(y ≯ x) for some x, y ∈ X. The former negation implies x > y while the latter implies y > x, a contradiction). Hence, Theorem 2 in Bergstrom et al (1976) is a restatement of Theorem 1 above.

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1 ⇒ 2. It is well-known that continuity of % implies hemicontinuity. The latter is equivalent to the openness of U% (w)c and L% (w)c for all w ∈ X. Let x 6% y. Since 6% is open, N (x, y) ⊂ % 6 for some neighborhood N (x, y) of (x, y) ∈ X ×X. From this it follows that there are neighborhoods N (x), N (y) of x, y in X such that N (x) × N (y) ⊂ N (x, y). Thus, x0 ∈ N (x) and y 0 ∈ N (y) implies x0 6% y 0 . Since 6% is irreflexive, N (x) ∩ N (y) = ∅. Suppose now that for z ∈ X, x 6% z and z 6% y. Since U% (z)c and L% (z)c are open and x ∈ U% (z)c , y ∈ L% (z)c , there are neighborhoods M (x) ⊂ U% (z)c and M (y) ⊂ L% (z)c . Let W (x) := M (x) ∩ N (x) and W (y) := M (y) ∩ N (y). From the above, W (x) ∩ W (y) = ∅ and W (x) × W (y) ⊂ % 6 . Hence, 6% expands local transitivity. 2 ⇒ 1. Suppose x 6% y. Let z ∈ X be such that x 6% z and z 6% y. Then, from the assumption that 6% expands local transitivity it follows that a neighborhood N (x, y) ⊂ 6% exists. If there is no z ∈ X such that x 6% z and z 6% y, then it holds that L% (x)c ∩ U% (y)c = ∅. By way of contradiction, assume that U% (y)c × L% (x)c 6⊂ % 6 . There exist then x0 ∈ c 0 c 0 0 0 U% (y) , y ∈ L% (x) such that x % y . Suppose x % x is also true. Then (x % x0 , x0 % y 0 ) ⇒ x % y 0 , which contradicts y 0 ∈ L% (x)c . Hence, x 6% x0 . But then x 6% x0 , x0 6% y holds. Hence, x0 ∈ L% (x)c ∩ U% (y)c . This contradicts the postulate L% (x)c ∩ U% (y)c = ∅. It follows therefore that U% (y)c × L% (x)c ⊂ 6%. Hemicontinuity implies that U% (y)c and L% (x)c are open, and hence that U% (y)c × L% (x)c is open. It has been shown therefore that in both cases an open neighborhood of (x, y) that is contained in 6% can be found, which proves that 6% is open, or equivalently, that % is continuous.  Example 2 Let X be a T2 space. The equality relation ‘=’ is a continuous preorder on X. Consider the inequality relation ‘6=’, which is irreflexive, symmetric and open in X × X. Suppose x 6= z, z 6= y and x 6= y. There exist N (x, z), N 0 (z, y) and N 00 (x, y) in X × X that are contained in 6=. Notice that Xz := X \ {z} is open because the space is also T1 , and that x, y ∈ Xz . Let N (x, z) ⊃ N (x) × N (z), N 0 (z, y) ⊃ N 0 (z) × N 0 (y) and N 00 (x, y) ⊃ N 00 (x) × N 00 (y), where N (·), N 0 (·) and N 00 (·) are neighborhoods in X. Clearly, N (x), N 0 (y) ⊂ Xz and N 00 (x) ∩ N 00 (y) = ∅. Let M (x) := N (x) ∩ N 00 (x) and M (y) := N 0 (y) ∩ N 00 (y). Then M (x), M (y) ⊂ Xz and M (x) × M (y) ⊂ = 6 , which shows that 6= expands local transitivity. ♦ Example 3 Let X = R be endowed with the co-finite topology T := {A ⊆ X : A = ∅ or |Ac | < ∞}. The space (X, T ) is T1 and the relation ‘=’ is a hemicontinuous preorder on X. It is not continuous, because (X, T ) is not T2 .8 Since no two points in X can be separated 8 See

Steen and Seebach (1995, p. 50).

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by elements of T , the relation 6= does not expand local transitivity.

♦

Remark: Proposition 1 in Nachbin (1950) states that a preorder % on a topological space X is continuous if and only if, for all x, y ∈ X such that x 6% y there exist disjoint open sets V 3 x and W 3 y such that V is %-decreasing and W is %-increasing.9 In view of Theorem 4, Nachbin’s condition is also equivalent to hemicontinuity of the preorder and expansion of local transitivity of its complement.  Since it is now clear that hemicontinuity without completeness is a primitive axiom for a preorder whereas continuity is not, it is naturally of interest to know whether there exists some behavioral property, which, when assumed together with hemicontinuity, implies that the weak preference relation’s complement expands local transitivity and therefore that the weak preference relation itself is continuous. It turns out that such a behavioral condition on the preorder’s complement does exist, but the predictions it generates are questionable. Definition 5 The relation 6% is distinctly transitive if x 6% y and y 6% z implies x 6% z for all distinct x, y, z ∈ X. Distinct transitivity of an irreflexive binary relation was studied in Harary (1961). In the present context, to require that 6% be distinctly transitive is equivalent to requiring % to be distinctly negatively transitive. The importance of the adjective “distinct” lies in the fact that an irreflexive and transitive relation is necessarily asymmetric, whereas an irreflexive and distinctly transitive relation need not be. In particular, if x, y ∈ X are such that x 6% y and y 6% x (i.e. if 6% is not asymmetric), transitivity of 6% would imply x 6% x, which obviously contradicts the reflexivity of %. Therefore, a preorder % cannot be negatively transitive unless it is complete. On the other hand, if 6% is merely distinctly transitive, then the above contradiction is avoided and % may well be incomplete. Indeed, it is not difficult to construct examples of incomplete preorders on finite sets whose complements are distinctly transitive relations. Clearly, if % is a hemicontinuous preorder on a topological space X and 6% is distinctly transitive, then 6% expands local transitivity. Thus, under %-hemicontinuity, distinct transitivity of 6% is a sufficient condition for % to be continuous. An example with the usual partial ordering, however, can easily demonstrate that it is not necessary. Nevertheless, this condition is arguably behavioral. Indeed, given distinct elements x, y, z ∈ X, one observes that the following are true when 6% has this property: 1. x f y and y f z implies x f z; 2. x f y and y  z implies x  z; 3. x  y and y f z implies x  z. According to the first implication, incomparability is a distinctly transitive relation. The second and third suggest that preference dominates incomparability in all relevant threealternative pairwise choice situations. Consider, however, the preferences of a single decision maker, who is cautious in the following sense: She is able to make preference comparisons if and only if both alternatives are familiar to her. Then, x  y implies 9 A set A ⊂ X is %-decreasing if x ∈ A, z ∈ X and x % z implies z ∈ A, and %-increasing if y ∈ A, w ∈ X and w % y implies w ∈ A.

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familiarity with both x and y, whereas, in view of this fact, y f z necessarily implies unfamiliarity with z. Hence, this decision maker should be unable to compare x and z too, but the prediction here is that x  z. Likewise, if she is unfamiliar with y but not with x and z, then x f y and y f z would both hold but x % z or z % x should also be true, in violation of x f z that is actually implied by distinct transitivity of f. To conclude this section, notice that the equivalence between statements 3 and 4 of Theorem 1 is included as a special case of Theorem 4 when the preorder % is complete. Indeed, since the complement of a hemicontinuous complete preorder is the inverse of the preorder’s asymmetric part (i.e. a hemicontinuous strict partial order), it obviously expands local transitivity.

4

(Hemi)Continuous Strict Partial Orders

Turning to incomplete strict preferences, one notes first that the kind of counterintuitive predictions of Corollary 3 that are valid for preorders and which are related to fragility are obviously ruled out in the present context once the preference relation is assumed continuous. Furthermore, given that hemicontinuity is generally primitive in this context as well, whereas continuity is not,10 a natural question is whether the latter follows from the former once some additional and behaviorally relevant assumption has also been made. It is well-known that the answer to this question is “yes”: By using the strict preference relation’s order-denseness property that is implied by standard non-satiation, Schmeidler (1969) showed that, in suitable domains, the latter suffices. Bergstrom et al (1976) used order-denseness directly to show in a simple way that the result is true for arbitrary spaces: Theorem 5 An order-dense strict partial order on a topological space is hemicontinuous if and only if it is continuous. In addition to non-satiation that is relevant in consumer-theoretic domains, in the context of preferences under risk or uncertainty where the domain is a convex subset of a real vector space, order-denseness of the strict preference relation  is also guaranteed not only by standard independence, but also by the much weaker betweenness axiom, which states that x  y implies x  αx + (1 − α)y  y for all α ∈ (0, 1). Finally, for X = Rn+ it was shown by Shafer (1974) that a hemicontinuous asymmetric relation (which need not be transitive) is continuous if it is convex in the standard consumertheoretic sense. For strict partial orders, therefore, the soundness of the behavioral origins of full continuity once hemicontinuity has been assumed is unambiguous. In light of the characterization of continuity in terms of hemicontinuity for possibly incomplete preorders that was provided in the previous section, it is natural to ask whether a similar result can be established for the case of strict partial orders. To provide a (hemi)continuity characterization analogous to that given in Theorem 4, one first notes that if > is a strict partial order, then, since > is asymmetric and therefore also irreflexive, ≯ is a reflexive relation. This basic observation, together with 10 Similar to the case of preorders, mere hemicontinuity suffices for an order topology to be generated on the domain of the strict partial order.

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Definition 4 that introduced expansion of local transitivity for a preorder’s complement, provides some motivation for the following topological condition which is to be satisfied by a strict partial order’s complement. Definition 6 The relation ≯ is continuously locally transitive if for all sequences (xn ), (yn ) and (zn ) with xn → x, yn → y, zn → z and also such that xn ≯ yn , yn ≯ zn , xn ≯ zn for all n ∈ N, it holds that x ≯ y, y ≯ z and x ≯ z. One notes that it does not follow directly from this definition that the relation ≯ is closed in X × X, although if ≯ is closed (i.e. if > is continuous) then ≯ clearly is continuously locally transitive. The theorem below shows that, in suitably restricted domains, continuity of a strict partial order is jointly characterized by hemicontinuity and continuous local transitivity of its complement. Theorem 6 The following are equivalent for a strict partial order > on a compact, connected, firstcountable topological space X: 1. > is continuous. 2. > is hemicontinuous and ≯ is continuously locally transitive. Proof: 1 ⇒ 2 is trivial. For the converse, assume, by way of contradiction that > is not continuous. There are sequences (xn ), (yn ) in X such that xn ≯ yn for all n ∈ N, xn → x, yn → y and x > y. Suppose L> (xn0 )c ∩ U> (yn0 )c = ∅ for some n0 ∈ N. Then L> (xn0 ) ∪ U> (yn0 ) = X.

(1)

Since xn0 ≯ yn0 , transitivity of > and (1) together imply L> (xn0 ) ∩ U> (yn0 ) = ∅.

(2)

Hemicontinuity, (1) and (2) then imply that X can be written as the union of two disjoint open sets, which violates connectedness. Hence, for all n ∈ N, L> (xn )c ∩ U> (yn )c 6= ∅.

(3)

Let a sequence (zn ) be defined by zn ∈ L> (xn )c ∩ U> (yn )c for all n ∈ N. In view of (3), (zn ) is well-defined. Since X is compact, there exists a subsequence (znr ) ⊆ (zn ) such that znr → z ∈ X. By construction, xnr ≯ znr , znr ≯ ynr and xnr ≯ ynr for all nr ∈ N. Clearly, xnr → x and ynr → y. Finally, since ≯ is continuously locally transitive, it also follows that x ≯ z, z ≯ y and x ≯ y. The latter part contradicts the initial postulate x > y.  Example 4 Let X = Rn and define > by x > y if xi > y i for all i ≤ n. It is easy to check that all three topological conditions above are satisfied by > and ≯, respectively. ♦

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Example 5 Let X = Rn and define > by x > y if xi ≥ y i for all i ≤ n and x 6= y. This relation is a strict partial order, but is not hemicontinuous: If x, y ∈ Rn are such that xi = y i for all i ≤ n − 1 and xn > y n , then x ∈ U> (y) but for every  > 0 the Euclidean open ball B (x) is not contained in U> (y). Furthermore, ≯ is not continuously locally transitive either: Let xm , ym , zm be the m-th terms of sequences in Rn constructed as follows: i i xim = ym = zm for all i ≤ n − 2, n−1 n−1 n−1 xm = a + 1, ym = a, zm = a − 1, 2 3 1 n n = a + , zm =a+ . xnm = a + , ym m m m

Clearly, xm ≯ ym , ym ≯ zm and xm ≯ zm for all m ∈ N, while xm → x, ym → y and zm → z such that x > y, y > z and x > z. ♦

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