@ Applied General Topology c© Universidad Politécnica de Valencia Volume 13, no. 1, 2012 pp. 81-89 Topological conditions for the representation of preorders by continuous utilities E. Minguzzi ∗ Abstract We remove the Hausdorff condition from Levin’s theorem on the repre- sentation of preorders by families of continuous utilities. We compare some alternative topological assumptions in a Levin’s type theorem, and show that they are equivalent to a Polish space assumption. 2010 MSC: 54F05, 54E55, 54D50. Keywords: Preorder normality, utilities, preorder representations, k-spaces. 1. Introduction A topological preordered space is a triple (E, T ,≤), where (E, T ) is a topo- logical space endowed with a preorder ≤, that is, ≤ is a reflexive and transitive relation [18]. A function f : E → R is isotone if x ≤ y ⇒ f(x) ≤ f(y), and a utility if it is isotone and additionally “x ≤ y and y � x ⇒ f(x) < f(y)”. In this work we wish to establish sufficient topological conditions on (E, T ) for the representability of the preorder through the family U of continuous utility functions with value in [0,1]. That is, we look for topological conditions that imply the validity of the following property x ≤ y ⇔ ∀f ∈ U,f(x) ≤ f(y). Economists have long been interested in the representation of preorders by utility functions [4]. More recently, this mathematical problem has found ap- plication in other fields such as spacetime physics [16] and dynamical systems [1]. ∗This work has been partially supported by GNFM of INDAM and by FQXi. 82 E. Minguzzi To start with, it will be convenient to recall some notions from the theory of topological preordered spaces [18]. A semiclosed preordered space E is a topological preordered space such that, for every point x ∈ E, the increasing hull i(x) = {y ∈ E : x ≤ y} and the decreasing hull d(x) = {y : y ≤ x}, are closed. A closed preordered space E is a topological preordered space endowed with a closed preorder, that is, the graph G(≤) = {(x,y) : x ≤ y} is closed in the product topology on E × E. Let E be a topological preordered space. A subset S ⊂ E is called increasing if i(S) = S and decreasing if d(S) = S, where i(S) = ⋃ s∈S i(s) and analogously for d(S). A subset S ⊂ E is convex if it is the intersection of an increasing and a decreasing set, in which case we have S = i(S) ∩ d(S). A topological preordered space E is convex if for every x ∈ E, and open set O x, there are an open decreasing set U and an open increasing set V such that x ∈ U ∩ V ⊂ O. Notice that according to this terminology the statement “the topological preordered space E is convex” differs from the statement “the subset E is convex” (which is always true). The terminology is not uniform in the literature, for instance Lawson [14] calls strongly order convexity what we call convexity. The topological preordered space E is locally convex if for every point x ∈ E, the set of convex neighborhoods of x is a base for the neighborhoods system of x [18]. Clearly, convexity implies local convexity. A topological preordered space is a normally preordered space if it is semi- closed preordered and for every closed decreasing set A and closed increasing set B which are disjoint, it is possible to find an open decreasing set U and an open increasing set V which separate them, namely A ⊂ U, B ⊂ V , and U ∩ V = ∅. A regularly preordered space is a semiclosed preordered space such that if x /∈ B, where B is a closed increasing set, then there is an open decreasing set U x and an open increasing set V ⊃ B, such that U∩V = ∅, and analogously, a dual property must hold for y /∈ A where A is a closed decreasing set. We have the implications: normally preordered space ⇒ regularly preordered space ⇒ closed preordered space ⇒ semiclosed preordered space. For normally preordered spaces a natural generalization of Urysohn’s lemma holds [18, Theor. 1]: If A and B are respectively a closed decreasing set and a closed increasing set such that A ∩ B = ∅, then there is a continuous isotone function f : E → [0,1] such that A ⊂ f−1(0) and B ⊂ f−1(1). A trivial and well known consequence of this fact is (take A = d(y) and B = i(x) with x � y) Proposition 1.1. Let E be a normally preordered space and let I be the family of continuous isotone functions with value in [0,1], then (1.1) x ≤ y ⇔ ∀f ∈ I,f(x) ≤ f(y). This result almost solves our original problem but for the fact that the fam- ily of continuous utility functions is replaced by the larger family of continuous isotone functions. Moreover, we have still to identify some topological condi- tions on (E, T ) in order to guarantee that E is a normally preordered space. It Topological conditions for preorder representations 83 is worth noting that Eq. (1.1) is one of the two conditions which characterize the completely regularly preordered spaces [18]. Let us recall that a kω-space is a topological space characterized through the following property [9]: there is a countable (admissible) sequence Ki of compact sets such that ⋃∞ i=1 Ki = E and for every subset O ⊂ E, O is open if and only if O ∩ Ki is open in Ki for every i (here E is not required to be Hausdorff). Recently, the author proved the following results [17] Theorem 1.2. Every kω-space equipped with a closed preorder is a normally preordered space. Theorem 1.3. Every second countable regularly preordered space admits a countable continuous utility representation, that is, there is a countable set {fk} of continuous utility functions fk : E → [0,1] such that x ≤ y ⇔ ∀k,fk(x) ≤ fk(y). Using the previous results we obtain the following improvement of Levin’s theorem.1,2 [15] [4, Lemma 8.3.4] Corollary 1.4. Let (E, T ,≤) be a second-countable kω-space equipped with a closed preorder, then there is a countable family {uk} of continuous utility functions uk : E → [0,1] such that x ≤ y ⇔ ∀k, uk(x) ≤ uk(y). Proof. Every closed preordered kω-space is a normally preordered space (Theor. 1.2). Since E is a second countable regularly preordered space it admits a countable continuous utility representation (Theor. 1.3). � With respect to the references we have removed the Hausdorff condition.3 Another interesting improvement can be found in [5]. In the remainder of the work we wish to compare this result with other reformulations which use different topological assumptions. 1.1. Topological preliminaries. Since in this work we do not assume Haus- dorffness of E it is necessary to clarify that in our terminology a topological space is locally compact if every point admits a compact neighborhood. 1These references do not consider the representation problem but rather the existence of just one continuous utility. Nevertheless, the argument for the existence of the whole representation is contained at the end of the proof of [4, Lemma 8.3.4]. 2The result [8, Theor. 1] should not be confused with this one, since their definition of utility differs from our. That theorem can instead be deduced from the stronger theorem 1.2. Also note that in their proof they tacitly use a kω-space assumption which can nevertheless be justified. 3In [10] it was first suggested that the Hausdorff condition could be removed. This generalization is non trivial and requires some care in the reformulation and generalization of some extendibility results [17]. 84 E. Minguzzi Definition 1.5. A topological space (E, T ) is hemicompact if there is a count- able sequence Ki, called admissible, of compact sets such that every compact set is contained in some Ki (since points are compact we have ⋃∞ i=1 Ki = E, and without loss of generality we can assume Ki ⊂ Ki+1). The following facts are well known (Hausdorffness is not required). Every compact set is hemicompact and every hemicompact set is σ-compact. Ev- ery locally compact Lindelöf space is hemicompact, and every first countable hemicompact space is locally compact.4,5 Definition 1.6. A topological space E is a k-space if for every subset O ⊂ E, O is open if and only if, for every compact set K ⊂ E, O ∩ K is open in K. We remark that we use the definition given in [21] and so do not include Hausdorffness in the definition as done in [7, Cor. 3.3.19]. Every first countable or locally compact space is a k-space.6 Thus under second countability “hemicompact k-space” is equivalent to local compactness. It is easy to prove that an hemicompact k-space is a kω-space and the con- verse can be proved under T1 separability (see [20, Lemma 9.3]). Further, in an hemicompact k-space every admissible sequence Ki, Ki ⊂ Ki+1, in the sense of the hemicompact definition is also an admissible sequence in the sense of the kω-space definition. The mentioned results imply the chain of implications compact ⇒ hemicompact k-space ⇒ kω-space ⇒ σ-compact ⇒ Lindelöf and the fact that local compactness makes the last four properties coincide. A continuous function f : X → Y between topological spaces is said to be a quasi-homeomorphism if the following conditions are satisfied [11, 6]: (i) For any closed set C in X, f−1(f(C)) = C. (ii) For any closed set F in Y , f(f−1(F)) = F . Every quasi-homeomorphism establishes a bijective correspondence ψf : CL(Y ) → CL(X) between the closed sets of Y and X through the definition ψf (C) = f−1(C). Remark 1.7. If f is surjective (ii) is satisfied. Furthermore, a quotient (hence surjective) map which satisfies f−1(f(C)) = C for every closed set C (or equiv- alently, for every open set) is a quasi-homeomorphism. Indeed, if C is closed then f(C) is closed, because of the identity f−1(f(C)) = C and the definition of quotient topology. Thus both properties (i)-(ii) hold, and f is a quasi- homeomorphism. The given argument also shows that f is closed (and open). Furthermore, it can be shown that a quasi-homeomorphism is surjective if and only if it is closed, if and only if it is open [6, Prop. 2.4]. 4In order to prove the last claim, modify slightly the proof given in [2, p. 486] replacing “Suppose no neighborhood Vi has a compact closure” with “Suppose x has no compact neighborhood”. 5A first countable Hausdorff hemicompact k-space space need not be second countable. Indeed, as stressed in [9] not even compactness is sufficient as the unit square with a suitable topology provides a counterexample [19, p. 73]. 6Modify slightly the proof in [21, Theor. 43.9] Topological conditions for preorder representations 85 2. Ordered quotient and local convexity On a topological preordered space E the relation ∼, defined by x ∼ y if x ≤ y and y ≤ x, is an equivalence relation. Let E/∼ be the quotient space, T / ∼ the quotient topology, and let � be defined by, [x] � [y] if x ≤ y for some representatives (with some abuse of notation we shall denote with [x] both a subset of E and a point on E/∼). The quotient preorder is by construction an order. The triple (E/ ∼, T / ∼,�) is a topological ordered space and π : E → E/∼ is the continuous quotient projection. Remark 2.1. Taking into account the definition of quotient topology we have that every open (closed) increasing set on E projects to an open (resp. closed) increasing set on E/∼ and all the latter sets can be regarded as such projec- tions. The same holds replacing increasing by decreasing. As a consequence, (E, T ,≤) is a normally preordered space (semiclosed preordered space, regu- larly preordered space) if and only if (E/∼, T /∼,�) is a normally ordered space (resp. semiclosed ordered space, regularly ordered space). The effect of the quotient π : E → E/∼ on the topological preordered properties has been studied in [13]. Remark 2.2. A set S ⊂ E is convex if and only if π(S) is convex. Indeed, let U and V be respectively decreasing and increasing sets, we have π(U ∩ V ) = π(U) ∩ π(V ) because: U ∩ V ⊂ π−1(π(U ∩ V )) ⊂ π−1(π(U) ∩ π(V )) = π−1(π(U)) ∩ π−1(π(V )) = U ∩ V . Proposition 2.3. Let (E, T ,≤) be a topological preordered space. If local convexity holds at x ∈ E then [x] is compact and every open neighborhood of x is also an open neighborhood of [x]. If E is locally convex then every open set is saturated with respect to π (that is π−1(π(O)) = O for every open set O). Hence π is a (surjective) quasi-homeomorphism, in particular π is open, closed and proper. Proof. Let O be an open neighborhood of x and let C be a convex set such that x ∈ C ⊂ O, then [x] = d(x) ∩ i(x) ⊂ d(C) ∩ i(C) = C ⊂ O, thus O is also an open neighborhood for [x]. The compactness of [x] follows easily. Let O ⊂ E be an open set and let x ∈ O. We have already proved that [x] ⊂ O. Since this is true for every x ∈ O, we have π−1(π(O)) = O. Therefore, by remark 1.7, since π is a quotient map it is a quasi-homeomorphism which is open and closed. Every such map is easily seen to be proper. � Remark 2.4. By the previous result under local convexity the quotient π es- tablishes a bijection between the respective families in E and E/∼ of open sets, closed sets, compact sets, increasing sets, decreasing sets and convex sets. Continuous isotone functions on E pass to the quotient on E/∼ and conversely, continuous isotone functions on E/∼ can be lifted to continuous isotone func- tions on E. As a consequence, many properties are shared between E and E/∼ regarded as topological preordered spaces (one should not apply this observa- tion carelessly, otherwise one would conclude that ≤ is an order and that T is Hausdorff). For instance, we have 86 E. Minguzzi Proposition 2.5. If E is a locally convex closed preordered space then E/∼ is a locally convex closed ordered space. Proof. We just prove closure to show how the argument works. If [x] � [y] then x � y. The representatives x and y are separated by open sets [18, Prop. 1, Chap. 1] Ux and Uy such that i(Ux) ∩ d(Uy) = ∅. By local convexity the increasing neighborhood of x, i(Ux), projects into an increasing neighborhood π(i(Ux)) of [x]. Analogously, π(d(Uy)) is a decreasing neighborhood of [y] which is disjoint from π(i(Ux)). We conclude that � is closed [18, Prop. 1, Chap. 1]. � The property of closure for the graph of the preorder does not pass to the quotient without additional assumptions [13]. For instance, the previous result holds with “locally convex” replaced by kω-space [17]. Remark 2.6. In a topological space (E, T ) the specialization preorder is defined by x � y if x ⊂ y. Two points x,y are indistinguishable according to the topology if x � y and y � x, denoted x � y, since in this case they have the same neighborhoods. The quotient under � of the topological space is called Kolmogorov quotient or T0-identification and gives a T0-space, sometimes called the T0-reflection of E. The Kolmogorov quotient is by construction open, closed and a quasi-homeomorphism. The first statement of proposition 2.3 implies that under local convexity if x ∼ y then x and y have the same neighborhoods, that is, x � y. If the preorder ≤ on E is semiclosed the converse holds because y = x ⊂ i(x) ∩ d(x), which implies, y ∼ x. Thus in a locally convex semiclosed preordered space, π is the Kolmogorov quotient and E/∼ is the T0-identified space. Actually, E/∼ is a T1-space because it is a semiclosed ordered space (remark 2.1) thus [x] ⊂ iE/∼([x]) ∩ dE/∼([x]) = {[x]}. If additionally E is a closed preordered space we already know that E/∼ is a closed ordered space. Another way to prove that � is closed is to observe that the T0-reflection of a product is the product of the T0-reflections, that is, π × π is the Kolmogorov quotient of E × E, and since the Kolmogorov quotient is closed it sends the closed graph G(≤) into the graph G(�) which is therefore closed. In summary we have proved Proposition 2.7. Let E be a locally convex semiclosed preordered space then π : E → E/∼ is the T0-identification of E and E/∼ is T1. Furthermore, if E is also a closed preordered space then E/∼ is a closed ordered space and hence T2. The next proposition will be useful (see Prop. 3.1) and is an immediate corollary of remark 2.4. Proposition 2.8. If (E, T ,≤) is (locally) convex then (E/ ∼, T / ∼,�) is (resp. locally) convex. If (E, T ,≤) is locally convex and locally compact then (E/∼, T /∼,�) is locally compact, and if additionally E is a closed preordered space then every point of E admits a base of closed compact neighborhoods (but E need not be T1). Topological conditions for preorder representations 87 3. Equivalence of some topological assumptions We wish to clarify the relative strength of some topological conditions that can be used in a Levin’s type theorem. Let us recall that a Polish space is a topological space which is homeomorphic to a separable complete metric space [3, Part II, Chap. IX, Sect. 6]. A pseudo- metric is a metric for which the condition d(x,y) = 0 ⇒ x = y, has been dropped [12]. The relation x ≈ y if d(x,y) = 0, is an equivalence relation and the quotient E/≈ is a metric space. A pseudo-metrizable space is a topological space with a topology which comes from some pseudo-metric. In particular, it is Hausdorff if and only if it is metrizable because the Hausdorff property holds if and only if the equiva- lence classes are trivial. We say that a space is a pseudo-Polish space if it is homeomorphic to a pseudo-metric space and the quotient under ≈ is a Polish space. Note that every pseudo-Polish space is separable. The next result is purely topological (see Prop. 2.7) but at some places it makes reference to a preorder. This is done because it is meant to clarify the topological conditions underlying a Levin’s type theorem in which the presence of a closed preorder is included in the assumptions. Proposition 3.1. Let us consider the following properties for a topological space (E, T ) and let ≤ be any preorder on E (e.g. the discrete-order) (i) second-countable kω-space, (ii) second-countable locally compact, (iii) pseudo-metrizable hemicompact k-space, (iv) locally compact pseudo-Polish space. Then (iv) ⇔ (iii) ⇒ (ii) ⇒ (i). Furthermore, if (E, T ,≤) is a locally convex semiclosed preordered space we have (i) ⇒ (ii), and if (E, T ,≤) is a locally convex closed preordered space we have (ii) ⇒ (iii) (note that the discrete-order is locally convex thus the former implication holds also under T1 separability of T and the latter implication holds also under T2 separability of T ). In par- ticular they are all equivalent if (E, T ,≤) is a locally convex closed preordered space (e.g. under Hausdorffness). Proof. We shall make extensive use of results recalled in the introduction. (ii) ⇒ (i). Every second countable locally compact space is an hemicompact k-space and hence a kω-space. (i) ⇒ (ii). Assume that (E, T ,≤) is a locally convex semiclosed preordered space. If we prove that E is hemicompact we have finished because first count- ability and hemicompactness imply local compactness. We have already proved that (E/∼, T /∼) is T1 (Prop. 2.7). But (E/∼, T /∼) is a kω-space by a non- Hausdorff generalization of Morita’s theorem [17] thus E/∼ is hemicompact [20, Lemma 9.3]. Let K̃i be an admissible sequence on E/∼, since π is proper (Prop. 2.3) the sets Ki = π −1(K̃i) are compact. They give an admissible sequence for the hemicompact property, indeed if K is any compact on E then 88 E. Minguzzi π(K) is compact on E/∼ thus there is some K̃i such that π(K) ⊂ K̃i. Finally, K ⊂ π−1(π(K)) ⊂ π−1(K̃i) = Ki. (ii) ⇒ (iii). A second countable locally compact space is an hemicompact k-space. Since (E, T ,≤) is a locally convex closed preordered space, E/∼ is Hausdorff (Prop. 2.7). Local convexity, local compactness, and second count- ability pass to the quotient E/∼ (see Prop. 2.3,2.8) which is therefore metriz- able by Urysohn’s theorem. Thus E is pseudo-metrizable with the pullback by π of the metric on E/∼. (iii) ⇒ (ii). A pseudo-metrizable space is second countable if and only if it is separable [12, Theor. 11 Chap. 4] thus it suffices to prove separability. In particular, since E is σ-compact it suffices to prove separability on each compact set Kn (of the hemicompact decomposition) with the induced topology (which comes from the induced pseudo-metric). It is known that every compact pseudo-metrizable space is second countable [12, Theor. 5, Chap. 5] and hence separable, thus we proved that E is second countable. As first countability and the hemicompact property imply local compactness we get the thesis. (iv) ⇒ (iii). E is a separable pseudo-metrizable space thus second countable [12, Theor. 11 Chap. 4]. Second countability and local compactness imply the hemicompact k-space property. (iii) ⇒ (iv). Since (iii) ⇒ (ii), E is second countable and locally com- pact. Let d be a compatible pseudo-metric on E and let E/ ≈ be the metric quotient. Since π≈ : E → E/≈ is an open continuous map (actually a quasi- homeomorphism) and E is second countable and locally compact then E/≈ is second countable and locally compact too. We conclude by [21, 23C] that the one point compactification of E/≈ is metrizable, and by compactness the one point compactification of E/ ≈ is completely metrizable. Further, since E/ ≈ is separable its one point compactification is also separable. We conclude that the one point compactification of E/≈ is Polish, and since E/≈ is Hausdorff and locally compact, E/≈ is an open subset of a Polish space hence Polish [3, Part II, Chap. IX, Sect. 6]. � Remark 3.2. 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Van Nostrand Company, Inc. (1965). [19] L. A. Steen and J. A. Seebach, Jr., Counterexamples in topology, New York: Holt, Rinehart and Winston, Inc. (1970). [20] N. E. Steenrod, A convenient category of topological spaces, Michigan Math. J. 14 (1967), 133–152. [21] S. Willard, General topology, Reading: Addison-Wesley Publishing Company (1970). (Received September 2011 – Accepted December 2011) E. Minguzzi (ettore.minguzzi@unifi.it) Dipartimento di Matematica Applicata “G. Sansone”, Università degli Studi di Firenze, Via S. Marta 3, I-50139 Firenze, Italy. Topological conditions for the representation of[4pt] preorders by continuous utilities. By E. Minguzzi