THE ROLE OF TOPOLOGY

THE ROLE OF TOPOLOGY

IN DECISION THEORY

J. C. Candeal1

1Universidad de Zaragoza

XVII Encuentro de Topologıa. Zaragoza, Noviembre 26-27,2010

J. C. Candeal Topology and Decision Theory

Outline

1 Introduction

2 Utility Theory: The Ordinal Approach

3 Utility Theory: The Topological Approach

4 Utility Theory: The Algebraic Approach

5 Social Choice Theory

6 Appendix

Outline

1 Introduction





6 Appendix

Outline

1 Introduction





6 Appendix

Outline

1 Introduction





6 Appendix

Outline

1 Introduction





6 Appendix

Outline

1 Introduction





6 Appendix

Outline

1 IntroductionMathematics and EconomicsDecision Theory





6 Appendix

IntroductionUtility Theory: The Ordinal Approach

Utility Theory: The Topological ApproachUtility Theory: The Algebraic Approach

Social Choice TheoryAppendix

Mathematics and EconomicsDecision Theory

Historical background

The marginalist period: 1838-1947

Cournot (theory of the firm and single market equilibrium, 1838),Walras (theory of the consumer and general equilibrium, 1874),Edgeworth (exchange economy and contract curve, 1881),Marshall (demand theory, 1890), Pareto (general equilibrium andoptimal resource allocation, 1896), Hicks (stability of equilibriumand barganing, 1946),...,etc.






Historical background

The marginalist period: 1838-1947

Cournot (theory of the firm and single market equilibrium, 1838),Walras (theory of the consumer and general equilibrium, 1874),Edgeworth (exchange economy and contract curve, 1881),Marshall (demand theory, 1890), Pareto (general equilibrium andoptimal resource allocation, 1896), Hicks (stability of equilibriumand barganing, 1946),...,etc.






Historical background (continued)

The set-theoretic/linear models period: 1848-1960

Arrow (social choice theory, 1951), Arrow-Debreu (generalequilibrium, 1954), McKenzie (general equilibrium, 1954), vonNeumann and Morgenstern (game theory, 1947), Nash (gametheory, 1950), Dantzig (linear programming, 1949),...,etc.







The set-theoretic/linear models period: 1848-1960

Arrow (social choice theory, 1951), Arrow-Debreu (generalequilibrium, 1954), McKenzie (general equilibrium, 1954), vonNeumann and Morgenstern (game theory, 1947), Nash (gametheory, 1950), Dantzig (linear programming, 1949),...,etc.







The period of integration: 1961-1980

Debreu (regular economies, 1970), Aumann (large economies,1964), Sen (social choice, 1970), Koopmans (optimal growththeory, 1965), Smale (global analysis, 1976),...,etc.

The period of spread: 1981-present

Decision theory (foundations, utility theory, risk, uncertainty,....,social choice), extensions of the Arrow-Debreu model (incompletemarkets, financial markets, infinite dimensional spaces,...),...,etc.


























Forewords

Utility theory: A term that formerly comes from economicsand related disciplines (also used in ethics, philosophy,psychology,..) and refers to a measure of the individual(collective) welfare (Jevons, Edgeworth, Pareto, Wold,...).

Confusion between preference and utility: “if a set of items isstrongly ordered, it is such that each item has a place of itsown in the order; it could, in principle, be given anumber”(Hicks (1956, p.19)).

Contributions to utility theory come from many disparatesources: pure maths (including operations research andstatistics), psychology (measurement theory), economics,....

Social choice theory: Is it possible to aggregate individualpreferences into a social one ?






Forewords










Forewords










Forewords










Forewords










Objectives

Goals:

Presenting in a unified framework both classical and recentresults in utility theory.

Introducing to social choice theory: combinatorial andtopological models.






Objectives

Goals:








Objectives

Goals:




Outline

1 Introduction

2 Utility Theory: The Ordinal ApproachThe Representation ProblemCharacterizing RepresentabilityThe Non-Representation Problem




6 Appendix




The Representation ProblemCharacterizing RepresentabilityThe Non-Representation Problem

The representation problem

Theorem (Cantor, 1895)

Let (X ,-) be a totally ordered set that is unbordered, dense, anddenumerable. Then there exists an order monomorphism from Xonto Q.

Let X be, P = {- ⊆ X × X ; - transitive and total} andS = {(a, b) ∈ R2; a ≤ b}

Question:

- ∈ P, ∃ u : X −→ R ; (x , y) ∈ - ⇐⇒ (u(x), u(y)) ∈ S ?










Question:

- ∈ P, ∃ u : X −→ R ; (x , y) ∈ - ⇐⇒ (u(x), u(y)) ∈ S ?










Question:

- ∈ P, ∃ u : X −→ R ; (x , y) ∈ - ⇐⇒ (u(x), u(y)) ∈ S ?










Question:

- ∈ P, ∃ u : X −→ R ; (x , y) ∈ - ⇐⇒ (u(x), u(y)) ∈ S ?






The representation problem (continued)

Notation. PR = {- ∈ P ;- is representable}

Definition

- ∈ P is perfectly separable if ∃ D ⊆ X , D countable in such away that for every x , y ∈ X , with (y , x) /∈ -, there is d ∈ D suchthat (x , d), (d , y) ∈ -.








Definition









Definition







Characterizing representability

Theorem (Milgram, 1941 and Birkhoff, 1948)

Let -∈ P. Then - ∈ PR iff - is perfectly separable.

Notation. For a given -∈ P, τ- denotes the order topology on X .

Theorem

Let -∈ P. Then - ∈ PR iff τ- is second countable.










Theorem











Theorem











Theorem







Set-up

Notation. Pa = {-∈ P; - is antisymmetric }.

Two canonical examples:

1. Lexicographic plane:

(R2,-L) :

(x1, x2) -L (y1, y2) ⇐⇒ x1 < y1, or x1 = y1 then x2 ≤ y2.

2. The first uncountable ordinal:

([0, ω1),-):

W - T ⇐⇒ W , T ∈ [0, ω1), W is order isomorphic to an ideal ofT .






Set-up




(R2,-L) :



([0, ω1),-):







Set-up




(R2,-L) :



([0, ω1),-):







Set-up




(R2,-L) :



([0, ω1),-):







Set-up




(R2,-L) :



([0, ω1),-):







Definitions and questions

Definition.

A chain (X ,-) is said to be:(i) planar (or, a Debreu chain) if it contains a subchain that isorder isomorphic to a non-representable subset of the lexicographicplane (R2,-L).

(ii) long if it, or its dual, contains a subchain which is orderisomorphic to [0, ω1).

Questions:

(1) Are there any other orders, different from the above two, thatwill cause problems ?

(2) How many such other orders can we find ?







Definition.



Questions:









Definition.



Questions:








Solving question (1):

Definition.

An uncountable short (i.e., no long) chain that does not containany uncountable representable subchain is called an Aronszajnchain.

Theorem (Beardon et al., 2002)

(i) There exists an Aronszajn chain.(ii) Every Aronszajn chain is order isomorphic to a subchain of(Qω1

00(−1, 1),-L).







Definition.




00(−1, 1),-L).







Definition.




00(−1, 1),-L).








Let (X ,-) be an arbitrary chain. Then the following assertions areequivalent:(i) - /∈ Pa

R .(ii) (X ,-) is long chain, a Debreu chain or an Aronszajn chain.

Remark.

A Souslin chain contains a copy of an Aronszajn chain.










Remark.











Remark.



Outline

1 Introduction


3 Utility Theory: The Topological ApproachThe Continuous Representation ProblemCharacterizing PCR

The Continuous Representability PropertyCharacterizing CRP



6 Appendix




The Continuous Representation ProblemCharacterizing PCRThe Continuous Representability PropertyCharacterizing CRP

The continuous representation problem

Let (X , τ) be a topological space, P the space of preferences andS = {(a, b) ∈ R2; a ≤ b}

Question:

- ∈ P, ∃ u : X −→ R, u continuous, such that(x , y) ∈ - ⇐⇒ (u(x), u(y)) ∈ S ?

Notation.

PC = {- ∈ P ;- is closed in X × X}

PCR = {- ∈ P ;- admits a continuous utility function}








Question:


Notation.










Question:


Notation.










Question:


Notation.










Question:


Notation.

PC = {- ∈ P ;- is closed in X × X}PCR = {- ∈ P ;- admits a continuous utility function}






Characterizing PCR

Theorem

PCR = PC⋂

PR .

Corollary (Debreu’s open-gap lemma)

If τ ≡ τ-, for some -∈ P, then -∈ PCR iff -∈ PR .






Characterizing PCR

Theorem

PCR = PC⋂

PR .








Characterizing PCR

Theorem

PCR = PC⋂

PR .








Two cornerstone results

Theorem (Eilenberg, 1941)

Let (X , τ) be a connected and separable topological space. Thenany continuous total preorder defined on X has a continuous utilityfunction.

Theorem (Debreu, 1964)

Let (X , τ) be a second countable topological space. Then anycontinuous total preorder defined on X has a continuous utilityfunction.


























CRP: Definition

Definition

Let (X , τ) be a topological space. Then τ satisfies the ContinuousRepresentability Property (CRP) if PC = PCR .






CRP: Definition

Definition

Let (X , τ) be a topological space. Then τ satisfies the ContinuousRepresentability Property (CRP) if PC = PCR .






Sufficient conditions for CRP

Theorem

Let (X , τ) be a topological space. Then CRP holds true in thefollowing cases:







Theorem

Let (X , τ) be a topological space. Then CRP holds true in thefollowing cases:







Theorem

Let (X , τ) be a topological space. Then CRP holds true in thefollowing cases:(i) If τ is connected and separable (Eilenberg, 1941).







Theorem

Let (X , τ) be a topological space. Then CRP holds true in thefollowing cases:(i) If τ is connected and separable (Eilenberg, 1941).(ii) If τ is second countable (Debreu, 1964).







Theorem

Let (X , τ) be a topological space. Then CRP holds true in thefollowing cases:(i) If τ is connected and separable (Eilenberg, 1941).(ii) If τ is second countable (Debreu, 1964).(iii) If τ is path-connected and σ-compact (Monteiro, 1987).







Theorem

Let (X , τ) be a topological space. Then CRP holds true in thefollowing cases:(i) If τ is connected and separable (Eilenberg, 1941).(ii) If τ is second countable (Debreu, 1964).(iii) If τ is path-connected and σ-compact (Monteiro, 1987).(iv) If τ is locally connected and separable (Candeal et al., 2004).







Theorem

Let (X , τ) be a topological space. Then CRP holds true in thefollowing cases:(i) If τ is connected and separable (Eilenberg, 1941).(ii) If τ is second countable (Debreu, 1964).(iii) If τ is path-connected and σ-compact (Monteiro, 1987).(iv) If τ is locally connected and separable (Candeal et al., 2004).(v) If τ is separably connected and satisfies CCC. In particular, thisimplies that the weak topology of a Banach space has CRP(Campion et al., 2006).






Some characterizations of CRP

Theorem

Let (X , τ) be a metric space. Then τ satisfies CRP iff it is secondcountable (≡separable).

Theorem

Let (X ,-) be a totally preordered space. Then τ- satisfies CRP iffit is second countable.







Theorem


Theorem








Theorem


Theorem







Some characterizations of CRP (continued)

Theorem

Let (V ,+, ·R, τ) be a locally convex topological real vector space.Then τ satisfies CRP iff it satisfies CCC.






Some characterizations of CRP (continued)

Theorem

Let (V ,+, ·R, τ) be a locally convex topological real vector space.Then τ satisfies CRP iff it satisfies CCC.






Characterizing CRP

Definition

A topology τ on X is said to be (totally) preorderable if τ ≡ τ-,for some -∈ P.

Theorem (Herden and Pallack, 2000 and Campion et al., 2002)

Let (X , τ) be a topological space. The topology τ satisfies CRP ifand only if all its preorderable subtopologies are second countable.






Characterizing CRP

Definition









Characterizing CRP

Definition





Outline

1 Introduction



4 Utility Theory: The Algebraic ApproachThe Case Of GroupsThe Case Of Real Vector Spaces


6 Appendix




The Case Of GroupsThe Case Of Real Vector Spaces

The additive representation problem

Let (X ,+) be a group, P the space of preferences andS = {(a, b) ∈ R2; a ≤ b}

Question:

- ∈ P, ∃ u : X −→ R, u additive, such that(x , y) ∈ - ⇐⇒ (u(x), u(y)) ∈ S ?

Definition

Let (X ,+) be a group. A total preorder -∈ P is said to betranslation-invariant if x - y =⇒ x + z - y + z andz + x - z + y , ∀x , y , z ∈ X .








Question:


Definition









Question:


Definition









Question:


Definition







Characterizing PAR

Notation. PA = {- ∈ P ;- is translation-invariant}.PAR = {- ∈ P ;- admits an additive utility function}.Remark. PAR PR

⋂PA.

Theorem

Let (X ,+) be a group and let -∈ P. Then the following assertionsare equivalent:..






Characterizing PAR

Notation. PA = {- ∈ P ;- is translation-invariant}.PAR = {- ∈ P ;- admits an additive utility function}.

Remark. PAR PR⋂

PA.

Theorem

Let (X ,+) be a group and let -∈ P. Then the following assertionsare equivalent:..






Characterizing PAR


⋂PA.

Theorem

Let (X ,+) be a group and let -∈ P. Then the following assertionsare equivalent:(i) -∈ PAR ..






Characterizing PAR


⋂PA.

Theorem

Let (X ,+) be a group and let -∈ P. Then the following assertionsare equivalent:(i) -∈ PAR .(ii) -∈ PA and there is a base of the origin for τ-, say Ve , suchthat, for each B ∈ Ve ,

⋃n∈N nB = X .






The continuous additive representation problem

Let (X ,+) be a group, τ a topology on X , P the space ofpreferences and S = {(a, b) ∈ R2; a ≤ b}

Question:

- ∈ P, ∃ u : X −→ R, u continuous and additive, such that(x , y) ∈ - ⇐⇒ (u(x), u(y)) ∈ S ?

Notation.PCAR = {- ∈ P ;- admits a continuous and additiveutility function}.








Question:










Question:










Question:








Characterizing PCAR

Theorem (De Miguel et al., 1998)

PCAR = PC⋂

PAR .

Corollary (algebraic version of the open-gap lemma for groups)

If τ ≡ τ-, for some -∈ P, then -∈ PCAR iff -∈ PAR .






Characterizing PCAR


PCAR = PC⋂

PAR .








Characterizing PCAR


PCAR = PC⋂

PAR .








The continuous algebraic representability property

Definition

Let (X ,+) be a group and τ a topology on X . Then τ satisfies theContinuous Algebraic Representability Property (CARP) ifPCAR = PC

⋂PA.

Theorem

Let (X ,+) be a group and τ a topology on X . Then the followingassertions are equivalent:(i) τ has CARP(ii) For each -∈ PC

⋂PA, there is an absorbing basis of the origin

for τ-.







Definition


⋂PA.

Theorem



for τ-.







Definition


⋂PA.

Theorem



for τ-.






A sufficient condition for CARP

Corollary

Let (X ,+) be a group and τ a connected topology on X . Then τsatisfies CARP.






The linear representation problem

Let (X ,+, ·R) be a real vector space, P the space of preferencesand S = {(a, b) ∈ R2; a ≤ b}

Question:

- ∈ P, ∃ u : X −→ R, u linear, such that(x , y) ∈ - ⇐⇒ (u(x), u(y)) ∈ S ?

Definition

Let (X ,+, ·R) be a real vector space. A total preorder -∈ P is saidto be homothetic if x - y =⇒ λx - λy , ∀ x , y ∈ X , λ ≥ 0.








Question:


Definition









Question:


Definition









Question:


Definition







Characterizing PLR

Notation. PH = {- ∈ P ;- is homothetic}.PL = PA

⋂PH .

PLR = {- ∈ P ;- admits a linear utility function}.

Theorem

Let (X ,+, ·R) be a real vector space and let -∈ P. Then thefollowing assertions are equivalent:






Characterizing PLR


⋂PH .


Theorem

Let (X ,+, ·R) be a real vector space and let -∈ P. Then thefollowing assertions are equivalent:






Characterizing PLR


⋂PH .


Theorem

Let (X ,+, ·R) be a real vector space and let -∈ P. Then thefollowing assertions are equivalent:(i) -∈ PLR .






Characterizing PLR


⋂PH .


Theorem

Let (X ,+, ·R) be a real vector space and let -∈ P. Then thefollowing assertions are equivalent:(i) -∈ PLR .(ii) -∈ PL and (X ,+, ·R, τ-) is a topological real vector space.






Characterizing PLR


⋂PH .


Theorem

Let (X ,+, ·R) be a real vector space and let -∈ P. Then thefollowing assertions are equivalent:(i) -∈ PLR .(ii) -∈ PL and (X ,+, ·R, τ-) is a topological real vector space.

Consequence. PLR = PL⋂

PR .






The continuous linear representation problem

Let (X ,+, ·R) be a real vector space, τ a topology on X , P thespace of preferences and S = {(a, b) ∈ R2; a ≤ b}

Question:

- ∈ P, ∃ u : X −→ R, u continuous and linear, such that(x , y) ∈ - ⇐⇒ (u(x), u(y)) ∈ S ?








Question:









Question:







Characterizing PCLR

Notation. Let (X ,+, ·R) be a real vector space equipped with atopology τ . PCLR = {- ∈ P ;- admits a linear and continuousutility function}.

Theorem

PCLR = PC⋂

PLR .

Corollary (algebraic version of the open-gap lemma for real vectorspaces)

If τ ≡ τ-, for some -∈ P, then -∈ PCLR iff -∈ PLR .






Characterizing PCLR


Theorem

PCLR = PC⋂

PLR .








Characterizing PCLR


Theorem

PCLR = PC⋂

PLR .








Characterizing PCLR


Theorem

PCLR = PC⋂

PLR .








An important consequence

Corollary

The only topological totally ordered real vector space is R or itsdual.







Definition

Let (X ,+, ·R) be a real vector space equipped with a topology τ .Then τ satisfies the Continuous Algebraic RepresentabilityProperty (CARP) if PCLR = PC

⋂PL.

Theorem

Let (X ,+, ·R) be a real vector space equipped with a topology τ .Then the following assertions are equivalent:(i) τ has CARP(ii) For each -∈ PC

⋂PL, (X ,+, ·R, τ-) is a topological real

vector space.







Definition


⋂PL.

Theorem



vector space.







Definition


⋂PL.

Theorem



vector space.







Corollary

Let (X ,+, ·R) be a real vector space equipped with a connectedtopology τ . Then τ satisfies CARP. In particular, if (X ,+, ·R, τ) isa topological real vector space, then τ satisfies CARP.

Remark

The analogous of the last statement of the corollary is not longerfor groups.







Corollary


Remark








Corollary


Remark



Outline

1 Introduction




5 Social Choice TheoryArrovian FrameworkTopological AggregationCharacterizating social welfare functionals

6 Appendix




Arrovian FrameworkTopological AggregationCharacterizating social welfare functionals

Arrow’s model

Definition

A social welfare functional F : Pn −→ P satisfies:

Pareto condition if, for any pair of alternatives x , y ∈ X andany preference profile (-j) ∈ Pn, we have that x -j y for allj ∈ N implies xF (-j)y , and also that x ≺j y for all j impliesxF (-j)sy .

the independence of irrelevant alternatives (IIA) condition if,for any pair of alternatives x , y ∈ X and any pair of preferenceprofiles (-j), (-

′j) ∈ Pn with the property that, for every j ,

x -j y ⇐⇒ x -′j y and y -j x ⇐⇒ y -

′j x , we have that

xF (-j)y ⇐⇒ xF (-′j)y and yF (-j)x ⇐⇒ yF (-

′j)x .






Arrow’s model

Definition





x -j y ⇐⇒ x -′j y and y -j x ⇐⇒ y -



′j)x .






Arrow’s model

Definition





x -j y ⇐⇒ x -′j y and y -j x ⇐⇒ y -



′j)x .






Arrow’s model (continued)

Theorem (Arrow, 1951)

Suppose that X contains at least three elements and letF : Pn → P be a social welfare functional that satisfies Pareto andIIA. Then there is an individual, say i , such that for every profile(-j) and every x , y ∈ X it holds that x ≺i y ⇒ xF (-j)sy .

Remark

Suppose that X contains at least three elements and letF : Pn

a → P be a social welfare functional that satisfies Pareto andIIA. Then there is an individual, say i , such that F (-j) =-i , forevery profile (-j) ∈ Pn

a .









Remark



a .









Remark



a .






Sketch of proof

Definition

A coalition A ⊆ N = {1, . . . , n} is said to be decisive if for any pairx , y ∈ X , any profile (-j) ∈ Pn such that x ≺j y , for every j ∈ A,then xF (-j)sy .

Fact (Fishburn, 1970 and Hansson, 1972)

Under the hypotheses of Arrow’s theorem the set of decisivecoalitions is an ultrafilter on N.






Sketch of proof

Definition









Sketch of proof

Definition









Chichilnisky’s model

Definition

Let (P, τ) be a topological space. A social choice ruleF : Pn −→ P is a map which satisfies:

anonymity, i.e., if the two profiles (pj), (qj) ∈ Pn are thesame up to a rearrangement, then F (pj) = F (qj).

unanimity, i.e., if F (p, p, . . . , p) = p, for every ∈ P.

continuity, i.e., if F is continuous

.







Definition





.







Definition





.







Definition





.







Definition




continuity, i.e., if F is continuous.






Some impossibility results in topological aggregation

Theorem

Suppose that P = S1, linear preferences over the closed disk B2,and let n = 2. Then there is no social choice rule.

Sketch of proof (Candeal and Indurain, 1994)

If such a social choice rule would exist then the boundary of theMobius strip would be a continuous deformation retract of thewhole strip. But this is not possible.







Theorem










Theorem









Some impossibility results in topological aggregation(continued)

Theorem

Suppose that P = S1, linear preferences over the closed disk B2,and let n ≥ 2. Then there is no social choice rule.

Sketch of proof

Based on Algebraic Topology techniques. If such a ruleF : (S1)n → S1 does exist then it can be shown that thefundamental group of S1, π1(S

1), is Abelian and any element isdivisible by n. But this leads to a contradiction since π1(S

1) = Z.







Theorem


Sketch of proof



1) = Z.







Theorem


Sketch of proof



1) = Z.






The resolution of the social choice paradox

Important facts

(1) Let P be a space of preferences with finitely generatedhomotopy groups. If a social choice rule does exist, for anyn ∈ N, then each homopoty group of P is trivial.

(2) Contractible spaces have the property that all theirhomopoty groups are trivial.

(3) Identify a class of topological spaces (P, τ) for which theconverse of (2) holds true.







Important facts










Important facts










Important facts










Important facts









The resolution of the social choice paradox (continued)

Theorem

Suppose that P is connected and homotopic to a polyhedron. If asocial choice rule exists for any number of individuals, then P iscontractible.







Theorem

Suppose that P is connected and homotopic to a polyhedron. If asocial choice rule exists for any number of individuals, then P iscontractible.







Question

When contractibility is a sufficient condition ?

Theorem

Suppose that P is connected and homeomorphic to a polyhedron.Then, a social choice rule exists for any number of individuals iff Pis contractible.

Theorem (Chichilnisky and Heal, 1983)

Suppose that P is a connected parafinite CW-complex. Then asocial choice rule exists for any number of individuals iff P iscontractible.







Question


Theorem










Question


Theorem










Question


Theorem









Further developments

(1) Dropping anonymity and replacing unanimity by Paretocondition while keeping continuity leads to the existence of aunique manipulator on spheres.

(2) Futher remarks and generalizations of Chichilnisky andHeal results can be given by Horvath (2001), Eckmann (2004),Weinberger (2004), Ardanza-Trevijano et al. (2007),. . . , etc.

(3) Arrow’s theorem can be formulated and proved using thetopological model. This fascinating link was discovered byBaryshnikov (1993) !
























Economics does not live by Topology alone

Definition

A social welfare functional F : Pn −→ P is said to be a two-sidedm serial dictatorship if there are a natural numberm ∈ N = {1, . . . , n}, an injection π of N(m) = {1, . . . ,m} into Nand a partition {A,B} of π(N(m)) such that, for any pair ofalternatives x , y ∈ X and any preference profile (-j) ∈ Pn, wehave that xF (-j)y if and only if x ∼π(k) y for all k ∈ N(m) orelse, there is i ∈ N(m) so that x ∼π(k) y for all k < j andx ≺π(i) y whenever i ∈ A, or y ≺π(i) x whenever i ∈ B.






A characterization result in the Arrovian model

Theorem

Suppose that X contains at least three elements and letF : Pn → P be a social welfare functional. Then the followingassertions are equivalent:(i) F satisfies PI and IIA,(ii) F is either trivial, or there is m ∈ N such that F is a two-sidedm serial dictatorship.

Proof

Based on theory of ordered algebraic systems.







Theorem


Proof








Theorem


Proof



Outline

1 Introduction





6 Appendix




Proof 1

Suppose that τ satisfies CCC and let - be a continuous totalpreorder defined on V . Observe that it is sufficient to show that itis representable. Since (V , τ) is a separably connected topologicalspace, by a result of Campion et al. (2006), there is a continuousorder-preserving function u : V → L, where L denotes the long line.Notice that u(V ) is an interval of L. Furthermore, there is anordinal α0 ∈ ω1 that bounds u(V ). Indeed, otherwise u(V )exhausts L and then by considering(Vα)α∈ω1 = (u−1(α, α + 1))α∈ω1 , we obtain an uncountable familyof pairwise disjoint non-empty open subset of V , which contradictsCCC. So, there is a countable ordinal that bounds u(V ) andtherefore u(V ) can be identified with a subset of the real line. Thismeans that - is representable.





Proof 1 (continued)

For the converse suppose that, by way of contradiction, (V , τ)does not satisfy CCC. Let us then show that there is a continuoustotal preorder defined on V which is not representable. This willimply that (V , τ) does not satisfy CRP. To that end, let (pβ)β∈I

be a family of seminorms that define the topology τ on V and let(Uα)α∈ω1 be an uncountable family of pairwise disjoint non-emptyopen subset of V . Without loss of generality we can assume that,for each α, there is εα > 0 such that Uα can be chosen of the formUα = {v ∈ V ; there are vα ∈ V , βα ∈ I and ; pβα(v − vα) < εα}.





Proof 1 (continued)

Indeed, we can single out an uncountable family of pairwisedisjoint non-empty basis open subset of V , say, (Vα)α∈ω1 . Recallthat each Vα is of the form Vα = {v ∈ V ; there are vα ∈ V ,β1, . . . , βn ∈ I and ε1, . . . , εn > 0 ; pβj

(v − vα) < εj ; j = 1, . . . , n}.Then, by considering for each α ∈ ω1, εα =min{εj ; j = 1, . . . , n}we can assume each Uα to be of the form described above.





Proof 1 (continued)

For each α ∈ ω1, consider the (non-trivial) real interval [0, εα].Notice that [0, εα] ⊂ R is order-isomorphic to [0, α] ⊂ L. Denotethis order isomorphism by φα. Define now the function u : V → Las follows:

u(v) ={φα(εα − pα(v − vα)), v ∈ Uα

0 , v ∈ V \⋃

α∈ω1Uα

It remains to prove that u so defined is a continuous function sincethen by considering the relation on V defined as: v -u w iffu(v) ≤ u(w), (v ,w ∈ V ), it is straightforward to see that -u is anon-representable continuous total preorder.





Proof 1 (continued)

To show the continuity of u let b ∈ L. Then it is sufficient to seethat u−1(L(b)) and u−1(G (b)) are open subsets of V . If b = 0then u−1(L(b)) = u−1(∅) = ∅ and u−1(G (b)) =

⋃α∈ω1

Uα both ofwhich are, obviously, open subsets of V .Now, if b ∈ L \ {0} then for each α ∈ ω1 such that b < α, let usdefine the subsets Aα = {v ∈ Uα; pα(v − vα) > εα − φ−1

α (b)} andBα = {v ∈ Uα; pα(v − vα) < εα − φ−1

α (b)}. Notice that both Aα

and Bα are non-empty open subsets of V . Then,u−1(L(b)) = (

⋃b<α Aα)

⋃(V \

⋃α∈ω1

Uα) and

u−1(G (b)) =⋃

b<α Bα, whence open subsets of V too. Thisconcludes the proof.





Proof 2

The “only if”part is obvious. For the converse, let u : G → R be anadditive order-preserving function for -. Let us show that u is alsocontinuous. To that end, recall that if S ⊆ R is a subgroup of theusual additive group of the reals, then S = {0}, or there is somea ∈ (0,∞); S = aZ = {az ; z ∈ Z}, or S is dense in R (see, e.g.Choquet (1966, p.56)).As in the proof of Theorem 4.1 it is sufficient to show that, forevery b ∈ R, u−1(L(b)) and u−1(G (b)) are open subsets of G .Consider the subgroup of the reals S = u(G ) ⊆ R. According tothe above result, three cases need to be distinguished. Case (i): IfS = {0} then the results is obvious because u is constant.





Proof 2 (continued)

Case (ii): If S = aZ, then there are uniques = u(g), s ′ = u(g ′) ∈ S such that b ∈ [s, s ′]. Thus,u−1(L(b)) = L(g ′) and u−1(G (b)) = G (g), which are open subsetsof G since u is order.preserving and - is continuous.Case (iii): If S is a dense subgroup of the reals thenu−1(L(b)) =

⋃{g∈G ;u(g)<b} L(g) and

u−1(G (b)) =⋃{g∈G ;b<u(g)} G (g), which are open subsets of G

since u is order.preserving and - is continuous. This ends theproof.





That’s all

THANKS





That’s all

THANKS


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