aa r X i v : . [ ec on . T H ] A p r Final topology for preference spaces
Pablo Schenone ∗ This version: April 7, 2020.First version: March 27, 2019
Abstract
Most decision problems can be understood as a mapping from a prefer-ence space into a set of outcomes. When preferences are representable viautility functions, this generates a mapping from a space of utility functionsinto outcomes. We say a model is continuous in utilities (resp., preferences)if small perturbations of utility functions (resp., preferences) generate smallchanges in outcomes. While similar, these two concepts are equivalent onlywhen the topology satisfies the following universal property: for each contin-uous mapping from preferences to outcomes there is a unique mapping fromutilities to outcomes that is faithful to the preference map and is continu-ous. The topologies that satisfy such a universal property are called finaltopologies. In this paper we analyze the properties of the final topology forpreference sets. This is of practical importance since most of the analysison continuity is done via utility functions and not the primitive preferencespace. Our results allow the researcher to extrapolate continuity in utilityto continuity in the underlying preferences.
Keywords: Decision Theory, topologyJEL classification: C02, D5, D01 ∗ I wish to thank Kim Border, Laura Doval, Federico Echenique, Mallesh Pai, Omer Tamuz,and participants of the LA theory fest for insightful discussions that helped shape this paper.All remaining errors are, of course, my own. Introduction
Consider the following results about a decision maker with Constant Elasticityof Substitution (CES) preferences. As the elasticity of substitution approacheszero, the CES function approaches a Leontief utility function. Likewise, as theelasticity of substitution approaches unity, the CES utility function approaches aCobb-Douglas utility function. Formally, the CES function is given by U p x , x ; ρ q “ p αx ´ ρ ` p ´ α q x ´ ρ q ´ ρ ,σ “ ` ρ , where σ is the elasticity of substitution and p x , x q P R . Then, we have thefollowing limits: U p x , x ; ρ q Ñ min t αx , p ´ α q x u , as ρ Ñ 8 U p x , x ; ρ q Ñ x α x ´ α , as ρ Ñ . It is natural to interpret these results in the following manner. As the elasticityof substitution converges to unity, the decision maker’s preferences approach thepreferences of a Cobb-Douglas decision maker. Thus, any result that is true fora Cobb-Douglas decision maker should be approximately true for a CES decisionmaker with σ «
1. For instance, a Cobb-Douglas consumer spends a constantfraction of his total expenditure on good 1 and spends the rest on good 2; thus,one expects a CES decision maker with σ « σ « any result that is true about Cobb-Douglas decision makers will be approximately true for a CES decision maker with σ «
1, then we need to preserve all continuous mappings from the utility space to any outcome space, not only the space of the demand functions.In this paper, we characterize the topology on preference spaces that is definedby the two properties above; such a topology is called the final topology on pref-erences. The final topology on preferences can be represented by the diagramin Figure 1. Consider any outcome space of interest—for example, the space ofMarshall/Hicks demand functions, indirect utility/expenditure functions, best re-sponse strategies in a game, etc.— and function g from preferences to the outcomespace of interest. The final topology satisfies that g is continuous if and only ifthere exists a unique continuous function, h , such that the diagram commutes;that is, h p u q “ p g ˝ F qp u q for all utility functions u . What this guarantees is thatthe preference space preserves the structure of continuous mappings from util-ity functions to the economic space of interest (demand functions, indirect utilityfunctions, best responses in games, etc.). Specifically, setting the outcome spaceequal to the preference space and setting function g equal to the identity function,we find that the representation mapping, F , must be continuous.Space Of PreferencesSpace Of Utility Functions Outcome SpaceRepresentation Mapping, F h g Figure 1: The final topology on the preference space makes the diagram commute andpreserves all continuous mappings.
3n this framework, we provide three theorems that relate substantive economicassumptions a researcher can impose on preference spaces to the topological prop-erties of the preference space.Theorem 1 shows that when no restrictions are imposed on the preference space,this topology is the trivial topology: the only open sets are the empty set andthe full space. The driving force behind this results is that total indifference, i.e. , the preference where no alternative is strictly preferred to another, is partof the preference space. Since total indifference is a preference that generatesuninteresting economic predictions, it is customary to exclude it from the setof admissible preferences; in our case, this rules out the trivial topology as acharacterization of the final topology.Theorem 2 shows that if we focus on all preferences except total indifference,then the final topology is not trivial, it is path-connected, but it is not Hausdorff.By path-connected, we mean that given any two preferences, either one can becontinuously distorted into the other; by Hausdorff, we mean that for any twodistinct preferences, there is a neighborhood of the first and a neighborhood of thesecond such that these two neighborhoods are disjoint. While path-connectednessis an appealing property, failing the Hausdorff property is an unappealing propertyof the final topology (see Subsections 1.1 and 1.2 for discussions on the economicimplications of these properties). What prevents the space from being Hausdorffis the presence of non-singleton indifference curves.Theorem 3 characterizes the final topology for the space of strict preferences,that is, preferences where no two points are indifferent. We show that the spaceof strict preferences is a Hausdorff space that is totally path-disconnected. More-over, proposition 1 shows that the set of strict preferences is the largest set thatis both Hausdorff and includes all strict preferences. This result indicates thatpath-connectedness and the Hausdorff property are mutually exclusive propertiesfor preferences spaces, and they hold (or fail) depending on whether we allow indif-ferences. Therefore, assumptions on indifference curves have topological meaning.As such, the topological properties of preference spaces are not a mere technicalitybut carry substantial economic meaning and methodological restrictions.4mportantly, throughout this paper, we do not impose any topology on the spaceof alternatives. Sections 2 and 5 discuss why it might be desirable to conduct thisexercise without assuming a specific topology on the space of alternatives. How-ever, as discussed in Section 2, all the work performed on topologies for preferencespaces impose topological assumptions on the space of alternatives. Hence, Sec-tion 5 characterizes the final topology on preferences when we impose topologicalassumptions on the space of alternatives. Comparing the final topology on prefer-ences when we do and do not impose a topology on alternatives highlights the rolethat topological assumptions on the preference space play in shaping the topologyof preferences.In short, the paper has two main takeaways. First, there is a nontrivial trade-offbetween methodological generality and intuitive properties of the preference space.Allowing for indifference in the space of preferences is natural but leads to a pref-erence space that is not Hausdorff. However, recovering the Hausdorff propertycomes at the cost of restricting attention to strict preferences only and losing path-connectedness. Thus, there is a fundamental trade-off between methodologicalgenerality, the intuitive properties of the topology on preferences, and the struc-ture of indifference curves. Second, by comparing the topology on preferences bothwhen we do and do not impose topological assumptions on the alternative spacesheds light on how the topologies on alternatives impact the topologies on prefer-ences. A discussion of why one might, or might not, want to impose a topologyon the preference space is delayed until Sections 2 and 5.The rest of this paper is organized as follows. Sections 1.1 and 1.2 discusswhy the Hausdorff property and path connectedness are appealing properties fora preference space. Section 2 discusses our exercise in the context of the relatedliterature. Section 3 presents the model, and Section 4 shows our main results.Section 5 extends the model to allow for topologies on the set of alternatives.
Failure of the Hausdorff property means that a constant sequence of decisionsadmits a limit point that is different from the constant decision itself. This is anunnatural property that we wish our model to avoid: in general, if we observe a5equence of constant decisions and we ask ourselves what is the limiting decision,it should only be the constant decision itself.To illustrate the importance of the Hausdorff property, suppose that we line upa sequence of consumers deciding their demand functions, all of whom have thesame preferences, ą . Then, all these consumers have the same demand function, D . For instance, all consumers could be Cobb-Douglas and D “ p αp w x , p ´ α q p w x q ,where w is wealth and p p , p q is the price vector.If the Hausdorff property fails, then the sequence p ą , ą , ą , ... q has some ˆ ą asa limit point, where ˆ ą is different from ą . Furthermore, ˆ ą has as a demandfunction ˆ D that is different from D . As a consequence, the sequence of demandfunctions p D, D, D, ... q has ˆ D as a limit point, thus violating the intuitive rule thata sequence of constant decisions should converge only to the decision itself. A model’s continuity in utilities (or preference) is intimately tied to a comparativestatics question: if a decision maker’s preference was to change slightly, would hisdecisions change drastically? However, in order to ask this question, slight changesto a decision maker’s preference must exist. A corollary of Theorem 3 is that thespace of strict preferences is totally path-disconnected. Hence, each singleton isan open set, so no such thing as a “slight” change to a consumer’s preference ispossible.Another way to see this result is to consider the following related question.Assume we consider two distinct preferences, ą A and ą B , and suppose that underpreference ą A the model outputs prediction z p ą A q , whereas under preference ą B ,the model outputs prediction z p ą B q . If we continuously change preference ą A intopreference ą B , is it true that the model’s outcome continuously changes from z p ą A q to z p ą B q ? This is the basis for homotopy-based comparative statics exercises(see Shiomura [9], Borkovsky [2], or Eaves and Schmedders [6] for applications).However, in order to conduct this exercise, we need the space of preferences to bepath-connected. Hence, losing path-connectedness implies losing access to localcomparative statics tools in the space of preferences.6 Relation to existing literature
To the best of our knowledge, all work conducted on topologies for preferencespaces comes from the literature on general equilibrium. Debreu [4] proposes atopology on the space of continuous preferences that is based on the Hausdorffsemimetric; Hildebrand [7] provides generalizations on Debreu’s work. In theseworks, the space of alternatives, X , is endowed with an exogenous topology suchthat any continuous preference ą can be identified with a closed subset of X ˆ X .In Debreu’s work, X is assumed to be compact, so ą is a closed compact subsetof X ˆ X . By contrast, Hildebrand assumes X to be locally compact, so ą is compact in the one-point Alexandroff compactification of X . The objective ofthese assumptions is to associate each preference with a closed subset of a compactspace; therefore, one obtains a separable space by endowing the preference spacewith the Hausdorff semimetric. Because the space of preferences is given thetopology of the Hausdorff semimetric, Hildebrand calls it the topology of closedconvergence . Lastly, Kannai [8] takes the work of Debreu and analyzes the specialcase when preferences are continuous and monotone .Outside the realm of general equilibrium, Chambers, Echenique and Lambert[3] use tools from the above literature to address the following question. Supposeone observes a finite but large dataset generated by picking maximal elements outof a preference ą . Can one find a preference ˆ ą such that ˆ ą rationalizes observedchoices and ˆ ą is “close” to the true preference ą ? They provide a positive answerunder the topology of closed convergence when preferences are assumed to be locally strict (see Border and Segal [1] or Section 5 for a definition of locally strictpreference).Our exercise differs from the previous exercises in three ways.First, we wish to find a topology for preference spaces that is independent ofany exogenously imposed topology on X . Note that the study of topology is thestudy of continuous mappings; thus, choosing a topology for a set is equivalentto choosing a structure for continuous mappings involving the set. For decisionproblems, the minimal requirement should be that optimal binary decisions arecontinuous; that is, if x is strictly preferred to y and one perturbs y slightly,7hen x should remain preferred. The smallest topology with this property is thetopology generated by the upper and lower contour sets of the preference understudy. Therefore, any topology on X that makes optimal behavior continuouswill inherently depend on the preference under consideration: the topology on R that guarantees continuity of behavior for a lexicographic agent is fundamentallydifferent from the topology on R that guarantees continuous behavior for a Cobb-Douglas agent. Insisting on a preference-independent topology on X and excludingany preference that generates noncontinuous behavior with respect to such anarbitrarily chosen topology is unnatural in the context of decision making. Section5 argues that our approach of remaining agnostic about the topology on X isequivalent to endowing X with the topology generated by the upper and lowercontour sets and allowing the topology on X to vary as preferences vary. In short,assuming X is not exogenously endowed with a topology is equivalent to assumingthat all preferences are inherently continuous; some are just discontinuous relativeto additional assumptions on X .Second, because we do not impose a topology on X , continuity imposes no disci-pline in our model. Comparing our topology to the topology of closed convergencehighlights how the exogenous topology on X (together with the continuity restric-tion on preferences) shapes the topology on preference spaces. For example, themetrization aspect of preference is purely driven by the topological assumptionson X since preference spaces are generally not Hausdorff in the absence of an ex-ogenous topology for X . As previously discussed, this means that the topologicalproperties of preference spaces are driven by considerations that are not based onthe economic decisions being taken.Finally, the topology of closed convergence is motivated by attempting to adaptthe Hausdorff semimetric for closed sets to the preference space; that demandfunctions are upper hemicontinuous in preferences is proved as a theorem. Ourapproach is slightly different. In our motivation, that the optimizing behavior iscontinuous in preferences is part of the definition of our topology, not a result: wewant to guarantee that an economic model is continuous in preference if and only ifit is continuous in utilities, regardless of whether the Hausdorff semimetric achievesthis objective. Furthermore, preserving the structure of continuous mappings from8tilities to behavior includes, but need not be restricted to, maximizing behavior.For instance, if one was to write a behavioral model where agents do not fullyoptimize (for example, the satisficing model in the spirit of Simon [10]) and themodel is continuous in utility functions, then our topology demands that it shouldalso be continuous in preferences. This result validates the methodology of lookingat utility functions as a shortcut for analyzing continuity in preferences universallyacross applications, not just demand functions. Let X be a relevant choice space, U be the set of all utility functions on X , and P be the set of all binary relations on X that admit a utility representation. Thatis, for each ą P P , there exists a function u P U such that for all x, y P X , x ą y ifand only if u p x q ą u p y q . While most preference representation theorems hinge on X being a separable topological space, Echenique and Dubra [5] characterize theconditions under which a preference admits a utility representation in the absenceof topological assumptions on X . Let F : U Ñ P denote the mapping thattakes each utility function to the preference it represents; that is, for all x, y P X , u p x q ą u p y q if and only if xF p u q y . We call mapping F the representation mapping .Finally, let T U denote the pointwise topology on U . Our goal is to find a topology on P that preserves the structure of continuousmappings from U to any abstract set Z . If such a topology exists, then thecontinuous mappings from P to Z should be in one-to-one correspondence with thecontinuous mappings from U to Z . This requirement is captured by the definitionbelow. Definition 1. A final topology on P is a topology T P that satisfies the followinguniversal property: a function g : P Ñ Z is continuous if and only if there existsa unique continuous function h : U Ñ Z . Graphically, the final topology on P is the topology on P that makes the diagramin Figure 2 commute and preserves the structure of continuous mappings from util- We use this topology on U ˚ for simplicity of our characterization. Lemma 3 and Remark 2show how to generalize to any other topology on U ˚ . Z . The uniqueness condition guaranteesthat there is a bijection between the continuous functions g and the continuousfunctions h ; thus, our topology endows the preference space with exactly the samecontinuous functions as admitted by the utility space. PU Z F hg
Figure 2: T P makes the diagram commute and preserves continuous mappings. In this section, we present our three theorems. We briefly describe the theoremsand follow up with formal statements. Proofs are provided in the Appendix.
Theorem 1 shows that without any restrictions on P , topology T P must be trivial.That is, the only open sets are the empty set and P itself. The driving forcebehind this result is that total indifference, that is, the preference such that anytwo alternatives are indifferent, is an element of P . To see this result intuitively,consider any preference ą P P and any utility function that represents it, u P U . Wecan always flatten u into the constant function 0 in a way that is both continuousand preserves the underlying preference ą ; that is, we can find a sequence p u n q n P N such that each u n represents ą and u n Ñ . Since the representation mapping iscontinuous, the total indifference (which we denote as » ) belongs to the closure of t ą u . Therefore, the closure of any singleton contains the same element, » . Theonly way this is possible is if the topology is trivial: total indifference is the gluethat binds the whole space together into essentially a single point.10 heorem 1. Let T P denote the final topology on P . Then, T P “ tH , P u . Since total indifference is a nongeneric preference, it is natural to ask how The-orem 1 changes when we exclude this preference from P . To formally present Theorem 2, we first provide some important notation. First,let P ˚ “ P zt»u , where » is the total indifference preferences, i.e. , »“ X ˆ X .By analogy, let U ˚ “ t u P U : u is not constant u be endowed with the subspacetopology; that is, a set G is open in U ˚ if G “ G X U ˚ , where G is an openset in U . In what follows, let T P ˚ denote the final topology on P ˚ relative to U ˚ .Moreover, let A Ă X be any finite set and ą P P ˚ be such that for all x, y P A either x ą y or y ą x , i.e. , x y . Then, let B p ą , A q “ t ˆ ą : p@ x, y P A q x ą y ô x ˆ ą y u .The set B p ą , A q consists of all preferences that agree with ą on the finite set A of strictly ranked alternatives.Once we exclude total indifference from the set of preferences, Lemma 3 andTheorem 2 characterize a basis for T P . Indeed, each set B p ą , A q is a basis ele-ment; thus, spanning all combinations p ą , A q generates a basis for T P ˚ . However,this topology is not Hausdorff. Furthermore, Proposition 1 shows that unless weconsider only strict preferences, that is, preferences where no two alternatives areindifferent, T P can never be Hausdorff. Since, as argued in Section 1, the Hausdorffproperty is an intuitive property to ask for, we further restrict attention to strictpreferences.Lemma 3 proves that F is an open map independent of the topology imposedfor U . As such, one can characterize the basis elements of P ˚ by pushing forwardthe basis elements of U ˚ . Thus, we obtain a clear characterization for P ˚ . Theorem 2.
Let B “ t B p ą , A q : ą P P ˚ , A is finite u . Then, B is a basis for T P ˚ . Two corollaries follow from Theorem 2. First, if X has at least three elements,then T ˚ P is not Hausdorff. To illustrate this result, take any triplet of points x, y, z P X and consider a preference ą P U ˚ such that x ą y ą w . Consider any11 P F ´ p ą q and construct the following transformations: u n p z q “ $’’’&’’’% u p z q y Á zu p y q ` n u p z q x ą z ą yu p z q z Á x Note that F p u n q “ ą for each n and that F p lim t u n uq “ ˆ ą ‰ ą . By the continuityof F , this means ˆ ą P t ą u , where t ą u denotes the closure of the singleton t ą u .Hence, singletons are not sequentially closed, implying P ˚ is not Hausdorff.Second, provided that X has at least three elements, P ˚ is connected. Thisresult is easily demonstrated by contradiction. Suppose G and G disconnect P ˚ . Then, for some collections p ą i , t x i , y i uq i P I and p ą j , t x j , y j uq j P J , we have G “Y i B p ą i , t x i , y i uq and G Y j B p ą j , t x j , y j uq ; without loss of generality, assume that x i ą i y i and x j ą j y j for all i P I and j P J . Assume that there exist i P I and j P J such that t x i , y i u ‰ t x j , y j u . Construct ą as follows: for all z R t x i , y i x j , y j u , x i ą y i , x j ą y j , and z „ y j . Then, ą P G X G , a contradiction. Hence, G “ B p ą , t x, y uq for some p ą , t x, y uq and G “ B p ą , t x, y uq for some ą suchthat ą |t x, y u ‰ ą |t x, y u . Consider ˆ ą defined as follows: for all z, z ‰ x, y ; z ˆ „ z ˆ ą x „ y . Then, ˆ ą R G Y G , a contradiction. Thus, there cannot be sets G and G that disconnect P ˚ .Lemma 7 shows a stronger result: U ˚ is path-connected when X has at leastthree elements. Since P ˚ “ F p U ˚ q and F is continuous, P ˚ is path-connected. Asdiscussed in Section 1, this is an important property of the preference space. Theorem 3 shows that in the subspace of strict preferences, the topology identi-fied in Theorem 2 is Hausdorff but totally path-disconnected. Thus, Theorem 3presents a fundamental trade-off. While it is natural to consider nonstrict prefer-ences, this implies that any topology that universally preserves continuous map-pings cannot be Hausdorff; conversely, while the Hausdorff property is a naturalproperty to ask of a topology, it comes at the cost of considering only strict prefer-12nce spaces. A consequence of these observations is that topological conditions onpreference spaces carry substantive behavioral assumptions about the indifferencestructures allowed in the preference space.As before, to formally present Theorem 3, we provide some useful notation.Formally, let P s “ t ą P P : p@ x, y P X q x y u and U s “ t u P U : p@ x, y P X q u p x q ‰ u p y qu . Analogously, we let T P s denote the final topology on P s relativeto U s . Proposition 1.
Assume P s ‰ H . Let P be Hausdorff and P s Ă P . Then, P “ P s . Proposition 1 shows that the set of strict preferences is the largest set of prefer-ences that is both Hausdorff and includes all strict preferences. This characteristicis what motivates the restriction of P ˚ to P s . Theorem 3.
Let B “ t B p ą , A q : ą P P s , A is finite u . Then, B is a basis for T P s . Interestingly, the basis for T P s is the same as that for T P ˚ ; however, by excludingindifferences from the set of possible preferences, we can show that P s is Hausdorff,but we lose connectedness. To see that P s is Hausdorff, consider any preference ą P P s and let ą ´ be the opposite preference. That is, for all x, y P X , x ą y ô y ą ´ x . Note that ą ´ P P ˚ ; therefore, this preference is well defined. Then, P ˚ zt ą u “ Y x,y P X B p ą ´ , t x, y uq : indeed, if ˆ ą P P ˚ zt ą u , it must disagree with ą on some pair t x, y u and, thus, must agree with ą ´ on pair t x, y u . Since each B p ą ´ , t x, y uq is open, the complement of t ą u relative to P s is open, showing thatthe singletons are closed.However, P s fails to be path-connected. This is in contrast to P ˚ , which fails tobe Hausdorff but is path-connected. To see why P s is not connected, consider anypreference ą and any pair x, y P X . Then, we can write X “ B p ą , t x, y uq Y B p ą ´ , t x, y uq . Thus, we can express X as a disjoint union of two proper open sets,showing that X is disconnected. Lemma 8 provides a stronger result: P s is totallypath-disconnected. An easy way to see this is to assume that X is finite: asmentioned in the previous section, when X is finite, P s is a discreet space in thesense that the singletons are open. Therefore, P s is totally path-disconnected.13 Imposing topologies on X Section 4 presents results that do not presume any topology on the space of out-comes, Z , nor on the space of alternatives, X , because we are interested in pro-viding a universal methodology under which continuity in utilities is equivalent tocontinuity in preferences. Theorems 1 and 2 suggest that any such topology failsto be Hausdorff at this level of generality. This result motivates Theorem 3 andthe study of P s . However, one may ask if imposing topological assumptions on X might yield a topology on P or P ˚ that is Hausdorff, thus eliminating the need torestrict attention to P s .There are two routes to consider for imposing a topology on X . First, one canassume X comes endowed with an exogenously given topology; for example, if X “ R , one may endow X with the topology induced by the Euclidean distance.Then, to guarantee that utility representations exist, one restricts attention topreferences that are continuous with respect to the given topology on X , thatis, preferences such that the upper and lower contour sets are open sets in thetopology of X . Alternatively, one may assume X does not have an exogenouslygiven topology and, instead, impose that for each ą P P , the topology on X is thetopology generated by the upper and lower contour sets. This approach generatesa family of topological spaces, p X, T X p ą qq ą P P , where T X p ą q is the topology on X generated by the upper and lower contour sets.In what follows, we analyze the final topology on P ˚ when we do and do notendow X with an exogenous topology. X with an exogenous topology, T X Since none of our results relied on a topology for X , the final topology on X isstill the one defined in Theorem 2, and the space is still totally connected but notHausdorff. However, assuming that X is endowed with an exogenous topology T X restores the Hausdorff property for a strict superset of P s . Indeed, in line withBorder and Segal [1], we say a preference, ą , is locally strict if, for every pair p x, y q P X such that x „ y we can find a pair that is arbitrarily close to p x, y q ,where the ranking is strict. Formally, we have Definition 3 below.14 efinition 2. Let p X, T X q be a topological space, and let P be the set of all pref-erences defined on X that admit a utility representation. We say ą P P is locallystrict if the following holds: for each p x, y q P X ˆ X such that x Á y and for everyneighborhood V of p x, y q , there exists p x , y q P V such that x ą y . Remark 1.
Note that the definition of locally strict preference is effective onlywhen x „ y . Otherwise, x “ x and y “ y always satisfy the definition. Further-more, since » is clearly not locally strict, the set of locally strict preferences is asubset of P ˚ . Moreover, as previously mentioned, one should further restrict attention to con-tinuous preferences; that is, preferences whose upper and lower contour sets areopen in T X . This ensures that utility representations exist. For concreteness, wedefine continuity below. Definition 3.
Let p X, T X q be a topological space, and let ą be a preference on X .We say ą is continuous if for all y P X the following are true:1.- t x : y ą x u is open in T X .2.- t x : x ą y u is open in T X We denote the set of continuous and locally strict preferences as P cls Theorem 4 shows that the space of continuous and locally strict preferences isindeed Hausdorff. Proposition 1 showcases the connection between the topologi-cal properties of preference spaces and the economic assumptions we make aboutindifference: concretely, the Hausdorff property is incompatible with nonstrictpreferences. Theorem 4 reinterprets that result: Theorem 4 showcases the connec-tions between the topological properties of the alternatives space, the topologicalproperties of preference spaces, and the economic assumptions we make about in-difference. As before, the Hausdorff property holds for a preference space as longas the preference is strict, but only in a local sense. Clearly, the exact meaning of“local” is given by the topology the researcher imposes on X : preferences that arelocally strict in one topology might not be locally strict in another topology. Onemay worry that the topological properties of P ˚ might depend excessively on thearbitrary topology imposed on X , which motivates the results in the next section. Theorem 4.
The space P cls is Hausdorff. .2 Endowing X with the topology T X p ą q for each ą P P ˚ Contrary to the results in the previous subsection, one may not want to imposean exogenous topology on X . In contrast to the topology generated by lower andupper contour sets, exogenous topologies on X have no economic meaning becausethey are unrelated to any economic decision. However, if we consider the topologyon X induced by the upper and lower contour sets, then this topology is intimatelytied to the decision maker’s economic decisions. Indeed, if ą is a preference and T X p ą q is the topology generated by ą , this topology has a clear economic inter-pretation: a sequence of alternatives p x n q n P N approximates an alternative x if anytime x is worse than some z but better than some z . Then, eventually x n isalso worse than z and better than z . Formally, for all alternatives z, z such that z ą x ą z , the sequence p x n q n P N eventually satisfies z ą x n ą z . Topologies T X that are independent of preferences have no such economic interpretation; incontrast, topology T X p ą q has economic meaning since it depends only on how thedecision maker perceives the alternatives, and it makes optimal behavior contin-uous. For example, a consumer choosing a bundle in R who perceives the goodsas perfect substitutes has discontinuous demand selections if we impose the Eu-clidean topology on R ; however, the discontinuity disappears if we endow X withthe topology generated by upper and lower contour sets.To expand upon the above discussion, consider the following two examples. First,a consumer has to choose amongst bundles of two goods, so X “ R . This con-sumer has a satiation point such that consuming more than 10 units of either goodproduces no additional wellbeing. If we endow R with the Euclidean distance,then x “ p x , x q “ p , q and y “ p y , y q “ p , q are “far away”. How-ever, from the decision maker’s perspective, these two points are indeed very closesince consuming p x , x q or consuming p y , y q is indifferent. The large distancebetween bundle x and bundle y is driven purely by an assumption with no connec-tions to the decision maker’s economic problem. Second, consider a person withpreferences over money. Presumably, for this person, more money is better, and athousand dollars is very far from a million dollars. However, if a thousand dollarsis the minimal expenditure required to hit the consumer’s satiation point (for ex-ample, bundle p , q in the first example), then any amount of money above a16housand dollars is not intrinsically far from the thousand dollars itself. Whethera million dollars and a thousand dollars are close or far should not depend on thewhims of arithmetic but, rather, on what satisfaction a consumer can obtain fromthe additional money. In short, this section states that all preferences are inherently continuous; somejust happen to be discontinuous with respect to an exogenously given topology.Informally, Proposition 4 shows that the space P cls is not Hausdorff, implyingthat imposing an exogenous topology on X (as we did in the previous section) isa necessary and sufficient condition for recovering the Hausdorff property on P cls .The proof relies on Lemma 9: every preference that is not totally indifferent islocally strict when X is endowed with the topology generated by the upper andlower contour sets. Thus, the same construction we used to show that U ˚ is notHausdorff applies. Proposition 2.
Let P cls ˚ “ t ą : ą is locally strict when X has topology T X p ą qu .Then, P cls ˚ is not Hausdorff in the final topology. This result shows that the trade-off between the Hausdorff property and path-connectedness is unavoidable if the only topological assumption on X is that X is endowed with the topology T X p ą q for each ą P P . As mentioned in the intro-duction, there is a fundamental trade-off between methodological generality, theintuitive properties of the topology on preferences, and the structure of indifferencecurves. While it is natural to dismiss the topological properties of P as a mere tech-nicality, this paper shows that topological properties on the preference space havestrong connections to relevant economic problems. Path-connectedness is relevantfor comparative statics exercises, but it is incompatible with the Hausdorff prop-erty, which is a natural property for most economic applications. Furthermore,the Hausdorff property is incompatible with non-singleton indifference curves. To further drive the point, consider a decision maker that must choose two lotteries thatpay off an amount between ´ X “ ∆ pr´ , sq . Suppose we endow X with an L p norm. Then, the two lotteries l, q P X may be close or far away depending on the value of p we choose, which is an ad hoc decision that disregards the only economic consideration in thisproblem: how does the decision maker rank the lotteries in X . This problem is avoided if onesimply declares that the open sets are those generated by the upper and lower contour sets ofthe decision maker’s preferences. eferences [1] K. C. Border and U. Segal. Dynamic consistency implies approximately ex-pected utility preferences. Journal of Economic Theory , 63(2):170–188, 1994.[2] R. N. Borkovsky, U. Doraszelski, and Y. Kryukov. A user’s guide to solvingdynamic stochastic games using the homotopy method.
Operations Research ,58(4-part-2):1116–1132, 2010.[3] C. P. Chambers, F. Echenique, and N. Lambert. Recovering preferences fromfinite data. arXiv preprint arXiv:1909.05457 , 2019.[4] G. Debreu.
Neighboring economic agents . University of Calif., 1967.[5] J. Dubra and F. Echenique. A full characterization of representable prefer-ences.
Documento de Trabajo/FCS-DE; 12/00 , 2000.[6] B. C. Eaves and K. Schmedders. General equilibrium models and homotopymethods.
Journal of Economic Dynamics and Control , 23(9-10):1249–1279,1999.[7] W. Hildenbrand. On economies with many agents.
Journal of economictheory , 2(2):161–188, 1970.[8] Y. Kannai. Continuity properties of the core of a market.
Econometrica:Journal of the Econometric Society , pages 791–815, 1970.[9] T. Shiomura. On the hicksian laws of comparative statics for the hicksian case:the path-following approach using an alternative homotopy.
ComputationalEconomics , 12(1):25–33, 1998.[10] H. Simon and J. March.
Administrative behavior organization . New york: freePress, 1976.
A Proofs
A.1 Proof of Theorem 1
Definition 4.
The
Sierpi´nski space is the topological space defined by S “ pt , u , tH , t u , t , uuq . emma 1. Let Z be any topological space. If g : P Ñ Z is continuous, then g isconstant.Proof. Let Z be any topological space and g : P Ñ Z be continuous. Let h : U Ñ Z be defined by h “ p g ˝ F q . Let ą P P be any preference, u P F ´ p ą q be such that } u } ă
1, and u n “ n u for each n P N . Such representations alwaysexists because } arctan p¨q} ď T , continuity of g implies continuity of h . Therefore, g p„q “ h p q “ h p lim n Ñ8 t u n uq “ h p u q “ g p ą q , where is the constant function taking value 0.The third equality follows from F p u n q “ F p u q (so that h p u n q “ h p u q for each n ).Since ą was arbitrarily selected this proves that g is constant.The following proposition shows that, regardless of the topology imposed on U , P has the trivial topology. Key to this result is that „P P and t ą u for each ą P P . Theorem 5.
Let T P denote the final topology on P . Then, T P “ tH , P u Proof.
Let S be the Sierpi´nski space and g : P Ñ S be continuous. By Lemma 1, g is constant. Thus, there exists exactly two continuous functions from P to S :the function that takes value 1 always, and the function that takes value 0 always.Hence, T P “ tH , P u . A.2 Proof of Theorem 2 and connectedness results.
The next lemma characterizes the open sets in P as a function of the open sets in U . While continuity of F implies that F ´ p G q is open in U for each open G Ă P ,the converse is generally not true. However, the universal property on T impliesthat the converse is true for the final topology. Lemma 2.
For each G Ă P , if F ´ p G q is open, then G is open.Proof. Let S “ pt , u , tH , t , u , t uuq be the Sierpi`nski space. Fix G Ă P . Define ρ : P Ñ S as ρ p G A q “ ρ p G q “
1. Define h : U Ñ S as h p u q “ u P F ´ p G A q and F p u q “ u P F ´ p G q . Note that h “ ρ ˝ F . By the universal propertydefining T , ρ is continuous iff h is continuous. Since h ´ p q “ F ´ p G q and F ´ p G q is open in U by assumption, then h is continuous. Thus, ρ is continuous. Hence, ρ ´ p q “ G is open. 19e now show that F is an open map; that is, F p G q is open whenever G isopen. We provide two proofs of this result. The first proof is in Lemma 3 anddoes not rely on the topology of U ˚ , while the second one does. For Lemma 3we use the following notation: ˝ denotes the operation of function composition,and Sym p U ˚ q is the symmetry group of U ˚ , that is, the set of permutations on U ˚ endowed with the operation of function composition). The intuition behindthe proof of Lemma 3 is that P ˚ is the quotient of U under the group action thatassociates each u P U ˚ with the set of utility functions that represent F p u q . It iswell known that the projection on the quotient via a continuous group action isan open map, so F is an open map. Lemma 3.
Let G be an open set in U ˚ . Then, F p G q is open.Proof. Let A be the set of increasing functions from R to R that are continuous inthe standard topology of R . concretely, A “ t f f : R Ñ R f is continuous and strictly increasing u .Then p A, ˝q is a group that acts continuously on U ˚ in the following way: a : A Ñ Sym p U ˚ q is defined as a p f qp u q “ p f ˝ u q . Since each strictly increasing functionhas a strictly increasing inverse, then a is well defined. Also, for each f P A , a p f qp u q “ v if, and only if, F p u q “ F p v q . Now let G be an open set in U ˚ . Then,by Lemma 2 F p G q is open if, and only if, F ´ p F p G qq is open, so it only remainsto show that F ´ p F p G qq is open. Notice that F ´ p F p G qq “ Y f P A a p G q . Since a is bi-continuous and G is open, then a p G q is open for each G , making F ´ p F p G qq open. Remark 2.
Because F is open then we can characterize the topology on P ˚ viait’s basis elements. Indeed, if D is a basis for the topology on U ˚ then F p D q is abasis for the topology on P ˚ . When U ˚ is endowed with the topology of pointwiseconvergence (arguably, the most commonly used topology for applications), thenthis basis is easy to characterize explicitly. The next lemmas do that. For the next lemma we use the following notation: if A Ă X and ą P P ˚ we use ą | A to denote the restriction of ą to A. Formally, ą | A “ ą Xp A ˆ A q . Lemma 4.
Let ą P P ˚ and A Ă X be a finite set such that for all x, y P A x y .Then, B p ą , A q “ t ˆ ą : ˆ ą | A “ ą | A u is open. roof. Let ą and A be as in the statement of the lemma. First, consider the case A “ H . Then, for all ˆ ą P P we obtain ˆ ą | A “ H “ ą | A . Therefore B p ą , A q “ P ˚ ,which is open. Next, consider A “ t x u Ă X . Then, x x , a contradiction. Thus,without loss of generality, | A | ě
2. By Lemma 2, we need to show that F ´ p B p ą , A qq is open. Let u P F ´ p B p ą , A qq . Let ε “ min t| u p x q ´ u p y q| : x, y P A u ą A is finite. Then, B p u, ε q Ă F ´ p B p ą , A qq , where B p u, ε q is the ball of center u and radius ε . Thus, F ´ p B p ą , A qq is open.For the following Lemma, let D be basis of intervals for T U ˚ . That is, D P D if,and only if, the following holds: there exists a finite set of elements x , ..., x N P X and a finite set of intervals I , ..., I N Ă R such that for all u P D ˚ , u p x n q P I n foreach n P t , ..., N u . Lemma 5.
Let D P D ˚ . Then F p D q is open. That is, F p D q P T P ˚ .Proof. Let D be as in the statement of the Lemma, and t x , ..., x N u , t I , ...I N u be the corresponding points in X and intervals in R . The proof is divided in twocases. Case 1:
Assume that X Nn “ I n “ H . Then, there is a subset T Ă t , ..., N u suchthat I n X I k “ H for all n, k P T . Indeed, let I n “ p a n , b n q for each n . Let¯ a “ max t a n : 1 ď n ď N u , and b “ min t b n : 1 ď n ď N u . Then, ¯ a ě b, elseeach z P p ¯ a, b q satisfies z P X I n , a contradiction. Thus, at least two intervalsare disjoint. Let A “ t x t : t P T u . Then, for all u, v P D and all x, x P A , u p x q ą u p x q ô v p x q ą v p x q . Without loss of generality enumerate the set A “ t x t : t P T u in a monotone increasing way. That is, for all u P D , if k ą n then u p x k q ą u p x n q . Then, F p D q “ t ą P P : p@ k, n P T q k ą n ñ x k ą x n u .Thus, Lemma 4 implies F p D q is open. Case 2:
Assume that X n “ N I n ‰ H . Then, F p D q “ P ˚ . Indeed, let ą P P ˚ and u P U be a representation of ą . Let I “ X n “ N I n . Then, without loss ofgenerality u p x q P I for each x P X . Thus, u P D . Hence, ą “ F p u q P F p D q . Toconclude, because P P T P and F p D q “ P then F p D q P T P . Lemma 6.
Let F : U ˚ Ñ P ˚ be the representation map. Then, F is open.Proof. Let V be open in U ˚ ; that is, V P T U ˚ . Then, there exists a family of21ets t D i : i P I u such that V “ Y i D i , where I is an arbitrary index set. Then, F p V q “ Y i F p D i q . Lemma 5 shows that F p D i q is open for each i , thus F p V q isopen. This shows that F is an open map.For the next Lemma, we use the following notation: B p ą , A q “ t ˆ ą : ˆ ą | A “ ą | A u , where A Ă X is a finite set, and ą P P such that, for all x, y P A , x ‰ y . Theorem 6.
Let B “ t B p ą , A q : ą P P ˚ , A is finite u . Then, B is a basis for T P ˚ .Proof. By Lemmas 3 and 6, the image of D via F is a basis for T P ˚ . Since F p D q “ B , then B is a basis for T P ˚ .For the next lemma, we use the following notation. Let u P U and x P X ; we use u ´ x to denote the restriction of u to X zt x u . That is, u ´ x : X zt x u Ñ X . Similarly,if r P R , then p r, u ´ x q denote the function v : X Ñ R defined by v p x q “ r and v p y q “ u p y q for y ‰ x . Lastly, if r P R , we use boldface to denote the constantfuntion r : r p x q “ r for each x P X . This is the analogous notation one uses forfinitely dimensional vectors, and is convenient for out purposes. Lemma 7.
Assume X has at least three elements. Then, U ˚ is path connected.Proof. Let u, v P U ˚ . Pick x, y P X such that the following hold:1.- u p x q ‰ u p y q ,2.- pD c P X q such that v p x q ‰ v p c q , and c ‰ y Such x, y exist because X ě
3. We will continuously transform u into v , showingthat U ˚ is not only connected, but path-connected. We proceed in three steps. Step 1:
For each s P r , s define t as follows t p s qp z q “ $’’’&’’’% u p x q z “ x p ´ s q u p z q ` sv p z q z ‰ x, yu p y q z “ y
22e must check that for each s P r , s , t p s q is a non-constant function ( i.e. , t p s q P U ˚ ). Indeed, t p q “ u P U ˚ , t p s qp x q “ u p x q ‰ u p y q “ t p s qp y q for s ą w ” t p q . Step 2:
For each s P r , s define t as follows t p s qp z q “ $’’’&’’’% p ´ s q u p x q ` sv p x q z “ xw p z q z ‰ x, y p ´ s q u p y q ` sv p x q z “ y We must check that for each s P r , s , t p s q is a non-constant function ( i.e. , t p s q P U ˚ ). Indeed, t p q “ w P U ˚ . Moreover, t p qp z q “ v p z q if z ‰ y and t p qp y q “ v p x q . Since v p c q ‰ v p x q then t p q P U ˚ . Finally, if t p s qp¨q was aconstant function for some s P p , q then t p s qp x q “ t p s qp y q . Thus, p ´ s q u p x q ` sv p x q “ p ´ s q u p y q ` sv p x q ô u p x q “ u p y q , a contradiction. Thus, t p q P U ˚ . Let w ” t p q . Step 3:
For each s P r , s define t as follows t p s qp z q “ $&% w p z q z ‰ y p ´ s q w p z q ` sv p z q z “ y We must check that for each s P r , s , t p s q is a non-constant function ( i.e. , t p s q P U ˚ ). Indeed, t p q “ w P U ˚ . Moreover, t p q “ v P U ˚ . Finally, assume t p s q was constant for some s P p , q . Then, t p s qp y q “ t p s qp x q “ t p s qp c q , so p ´ s q v p x q ` sv p y q “ v p x q “ v p c q ô v p x q “ v p y q “ v p c q , a contradiction. Thus, t p s q P U ˚ for all s .Combining steps 1, 2, and 3 generates a continuous transformation of u to v withing U ˚ , and this concludes the proof. Remark 3.
Suppose X “ t x, y u ; then U ˚ “ tp u x , u y q P R : u x ‰ u y u . Define x “ tp u x , u y q P R : u x ą u y u and G y “ tp u x , u y q P R : u y ą u x u . Then, G x and G y disconnect U ˚ .A.3 Proof of Theorem 3 and Proposition 1 Theorem 7.
Let B “ t B p ą , A q : ą P P s , A is finite u . Then, B is a basis for T P s .Proof. The proof of Theorem 2 applies verbatim, since that proof never used thatpreferences where not strict. Indeed, the proof of Theorem 2 relies on » beingexcluded from P , which is still true in this case. Lemma 8. P s is totally path disconnected.Proof. It suffices to show that U s is totally path disconnected. Take u, v P U s be distinct functions. Assume there is a path t joining u and v . That is, t : S Ñ U s with t p q “ u , t p q “ v and t continuous. Since u ‰ v there are points x, y P X such that u p x q ą u p y q and v p y q ą v p x q . Let ∆ p s q “ t p s qp x q ´ t p s qp y q .Then, ∆ p q ą ą ∆ p q . Thus, there is a point s ˚ such that ∆ p s ˚ q “
0. Then, t p s ˚ qp x q “ t p s ˚ qp y q , contradicting that t p s ˚ q P U s . Thus, no two distinct functionsare path-connected. Proposition 3.
Assume P s ‰ H . Let P be Hausdorff and P s Ă P . Then, P “ P s .Proof. Assume not. Then, P is Hausdorff and there exists ą P P z P s . Let x, y P X such that x „ y . Such points must exists because ą R P s and assume that thisis the only instance of indifference. This assumption is without loss of generality. Furthermore, let A “ t z : x „ z „ y z ‰ x, z ‰ y u . If A ‰ H , let v : A Ñ r , s be a strict ranking of the points in A . That is, p@ a, b P A q , v p a q ‰ v p b q . Let u The construction below can trivially be replicated for all pairs p x, y q such that x „ y bylooking at the quotient of X ˆ X relative to indifference. Let ă x ą represent the indifferenceclass of x . For each class ă x ą that is not a singleton, pick two representatives, x, y , and carryout the construction below. ą . For each n P N , define the following utility functions: u n p z q “ $’’’&’’’% u p z q ´ n x ą z or z “ xu p z q ` n z ą y or z “ yu p x q ` v p z q z P A Notice the following: u n P P s Ă P for all n , u n Ñ u , F p u q “ F p u n q for all n P N , and F p ą q ‰ ą . Thus, ą P t F p u qu , thus violating that P is Hausdorff. A.4 Proof of Theorem 4
Theorem 8.
The space P cls is Hausdorff.Proof. Assume ą is locally constant and continuous. Assume, by way of contra-diction, that ˆ ą P t ą u , ˆ ą ‰ ą . Then, ˆ ą “ lim n Ñ8 ą n , where ą “ ą “ ... “ ą . ByTheorem 2, this means that if y ˆ ą x then y ą x . Thus, ˆ ą ‰ ą mens that there exitsa pair p x, y q P X ˆ X such that x ą y and x ˆ „ y . Because ą is continuous thismeans there are neighborhoods V x of x and V y of y such that x ą y for all x P V x and y P V y . Furthermore, because ˆ ą is locally strict and x ˆ Á y , then there exists x P V x and y inV y such that y ą x . Because ą n Ñ ˆ ą , y ˆ ą x implies y ą x .However, this contradicts that x ą y for all x P V x and y P V y . Lemma 9.
Let ą P P ˚ . Then, ą ˚ is locally strict in p X, T X p ą ˚ qq .Proof. Let p x, y q P X be such that x Á y . If x ą y then the locally strict propertyhold vacuously. Assume then that x „ y . Then, there wither exists z such that z ą x or there exists z such that y ą z , or both. Otherwise, for each z thefollowing holds: x Á z Á y „ x , implying that ą R P ˚ , a contradiction. Then,let V be a neighborhood of p x, y q . Then, there is a basis element A P T X p ą q and a basis element B P T X p ą q such that p x, y q P A ˆ B Ă V . By definition ofbasis elements, there exists z P A and z P B such that z ą x and y ą z . Thus, p z, z q P V and z ą z , proving that ą is locally strict. Proposition 4.
Let P cls ˚ “ t ą : ą is locally strict when X has topology T X p ą qu .Then, P cls ˚ is not Hausdorff in the final topology. roof. Because U ˚ is not Hausdorff, then ˆ ą P t ą u for distinct preferences ą , ˆ ą P P ˚ . Furthermore, Lemma 9 implies ˆ ą is locally strict in p X, T X p ˆ ą qq and ą islocally strict in p X, T X p ą qq . Thus, ą , ˆ ą P P cls ˚˚