In this note, we prove that for every non-separable metric space there is a continuous preference ordering which is non respresentable by an utility function.Preference Ordening; Utility Function; Non Separable Metric Space;

Arias de Reyna, Juan

Estévez Toranzo, Margarita

Hervés Beloso, Carlos

English

Research Papers in Economics

Working  paper  93-17  Departamento de  Economia 
Economy  Series  04  Universidad Carlos III de  Madrid 
September  1993  Calle Madrid,  126 
28903  Getafe  (Madrid) 
Fax  (34  1)  624-9875 
ON NON REPRESENTABLE PREFERENCES 
Juan Arias de  Reyna,  Margarita  Estevez  Toranzo  and  Carlos￿ 
Herves  Beloso·￿ 
Abstract  ---,  _ 
In this note,  we  prove that for every non-separable metric space 
there  is  a  continuous  preference  ordering  which  is  non 
representable  by  an utility function. 
Key  words 
Preference  Ordening;  utility  Function;  Non  Separable  Metric 
Space. 
·Arias,  Departamento  de  Analisis  Matematico,  Universidad  de 
Sevillaj  Estevez,  Departamento  de  Matematicas,  Universidad  de 
Vigoj  Herves,  Departamento  de  Economia,  Universidad  Carlos  III 
de  Madrid.  The  work  of  M.  Estevez  Toranzo  and  C.  Herves  Beloso 
is  partially  supported  by  Research  Grant  PS91-0042  from  the 
Direccion  General  de  la  Investigaci6n  Cientifica  y  Tecnica, 
Spanish Ministry of Education. 1 
1.  Introduction 
This work is concerned with the numerical representation of all continuous pref-
erence orderings on a topological space.  As  it is  well  known, if X  is  a cOllnected 
and separable topological space,  then continuous preference orderings on X  al-
ways have utility representations (see Eile-nberg (1941) and Debreu (1954». The .. 
assumption of connectedness is not necessary in the setting of metric spaces:  if X 
is perfectly separable, every continuous preference ordering is representable by an 
utility function (Debreu (1954»). 
However,  we  show here that separability is  also a  necessary condition for  the 
representability of all continuous preference orderings on a metric space. That is, 
if X  is a llon separable metric space, there exists a continuous preference ordering 
which does not admit an utility representation.  This is relevant since consumption 
sets in infinite dimensional commodity spaces are not separable, in general. 
2.￿  Definitions 
A preference ordering on the set X  is, to be precise, a binary relation on X, say 
~,  which is  reflexive, transitive and complete. 
An utility representation for  the preference ordering  ~  on X  is  a  function u : 
X  ~  R such that ;r  ~  y if and only if u(x) :5  u(y). 
Let  X  be a  topological  space.  We  say  that X  is  separable if it  contains  a 
countable subset whose  closure is  X.  We  say  that X  is  perfectly separable (or 
that X  satisfies the second countability axiom) if there is a countable class of open 
subsets such that everjr open subset in X  is  the union of sets of that class.  Every 
perfectly separable topological space is separable.  Every separable metric space is 
perfectly separable.  A topologlcal space X  is connected if there is no partition of 
X  into two disjoint, non-empty closed sets.  We  say  that X  is  path connected if 
for  all x,y in X  there is  a continuous function  f  : [0,.1]  ~  X  with f(O) = x  and 
f(l) = y.  Note that every path connected space is  connected and every convex 
set in a linear topological space is path connected. 
A preference ordering ~  on a topological space X  is continuous if the sets {x  E 
X  ; x  ~  x'},  {J~  EX; x' ~ x}  are closed for all x' E X. A subset B  C X  bounds 
~  if for every x  E X  there are a, b in B with a ::;  x  ~  b.  A preference ordering ~ 
is  countably bounded if there exists a countable set B C  X  that bounds::;.  Any 
preference ordering which has an utility representation is countably bounded. 2 
3.  The existence theorem 
THEOREM:  Let  X  be a  non separable metric space.  Then there is  a continuous 
preference ordering on X  which cannot be represented by an utility function. 
To prove the theorem we  shall make use of an auxiliar space L  called the long 
line (see Monteiro (1987), example 5, p.  151).  Let 0 1  be the least non-countable 
ordinal.  We denote by 0  the set of all ordinals et such that et < 0 1,  That is  to 
say that 0  is  the set of all countable ordinals.  Note that n is a well ordered set, 
non-counta.ble and such that for all 0  E 0, {,8 E 0  j  ,8  :5  et} is countable. 
Between each et  E 0  and its follower  et + 1 put one copy of the real interval 
(0,1).  The space L  that we  get, ordered in the obviuous way,  is called the long 
line.  We consider on L the order topology.  The details on the topological space L 
can be seen in Steen and Seebach (1970, pp.71,72). 
LEMMA: For each a E L, a :F 0 the order interval [0, a] = {x E L  j  0:5 x :5 a}  is a 
compact set homeomorphic to the real interval [0,1]. 
Proof.  It is  clear that it suffices  to  prove  the result  when a  = 0  E  O.  As 
{I' EO;  I'  :5  et}  is  a well  ordered countable set, there is  an order preserving 1 : 
{O, 1. ... ,o} -+ [0,1] such that f(O)  = 0 and I(et) = 1.  We define j : [0,0] -+ [0,1] 
by 
j(b) = I(b)  if bE 0  and 
j(b) = 1(1') + t(f(1J +1) - 1({3))  if  b =,8 +t,  ,8 EO, t E (0,1). 
It is  clear that j  is  an isomorphism of the order structures.￿ 
PROOF  OF  THE THEOREM: Let X  be a non separable metric space.  Non separable￿ 
metric spaces are ca.racterized by  the following  property:￿ 
There are c > 0 and an uncountable set D C X  such that 
for all x, yE D, x :F y implies d(x, y) ~ 3e.  (1 ) 
Otherwise, for  each € = ~,  n  EN, there exists a  co~ntable  set D  n  verifying (1) 
such that X =UaED  B(a, i), where  B(a,~) = {x E X,d(x,a) < '!;J.  Then the 
n 
set D = UDn  will be countable and dense. 
As  D  is  uncountable,  for  each  0  E  0  we  can choose  an x 0 E  D  in such  a 
way  that  0  '#  {3  implies xo':F  Xf3.  By  the lemma, for  each  et  E  0  there exist 
'Po : [0, e]  -+ L, which is an isomorphism between the order structures of [0. .::]  C R 
and [CPo(O), l'o(c)] =[0,0] C L 3￿ 
Let U : X  -+ L be defillf'd by 
o  if  x  ~ UoEo B(xo,e) U l'  - (  ) - {c.po(e:  - d(xo, x))  if  x E B(xo,e) 
It is  clear that U  is  continuous in B(x0' e) because c.po  and d are continuous.  If 
x  E  X  is  such  tha.t  d(;ro,.T)  = e,  we  have  U(x) =  0,  and c.po(O)  = 0,  then U 
is  continuous in x.  As  the intersection of two different  balls is  empty and U is 
constant in the exterior of UoEo B(xo , e), we have that U is continuous in X. 
For x,y in X, we  define x :$  y if and only if U(x) $  U(y).  It is clear that :$  is 
a  continuous preference ordering on X, but has no utility representa.tion because 
is not countably bounded.  To see it, note that given a countable set B  C X  there 
exists OB  E n such that sUPbEB U(b) < ° and then there is  not a  countable set 
B  C  X  that bounds :$. 
4.  Final remark 
\Ve remark that separability is not a necessary condition for the representability 
of all  preference orderings on  a  general  topological space X.  Monteiro (1987) 
proves that a continuous preference ordering on a path connected topological space 
X  is representable if and only if it is numerably bounded.  A continuous preference 
ordering on a compact topological space has one best and one worst point.  Then 
any  continuous  preference ordering on a  compact or O'-compact  (an union of a 
countable family of compact sets) path connected topological space is representable 
by utility functions.  Note that any compa.ct or O'-compact metric space is separable 
but compact topological spaces in general need not to be separable. 
--------------------' 4 
References 
[1]￿  Debreu, G; (1954).  Representation of  a Preference Ordering by a Numerical 
Function, in Decision Procesess, R.M. Thrall, C.H. Coombs and R.L. Davis 
eds., Wiley N.Y. pp 159-165 and also in Mathematical Economics:  Twenty 
papers of Gerard  Debreu.  Cambridge  University  Press.  Cambrid~e 
105-110. 
[2]￿  Eilemberg,S;  (1941).  Ordered  Topological  Spaces.  American  Journal of 
Mathematics, 63, 39-45. 
[3]￿  Monteiro, P.K. (1987).  Some results on the existence of utility functions on 
path connected spaces.  Journal of Mathematical Economics, 147-156. 
[4)￿  Steen, L.A.  and J.A. Seebach Jr.  (1979).  Counterexamples in topology. 
Springer Verloe:.  N.Y. 

On non representable preferences.

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On non representable preferences.

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