CaCeBuAGT.dvi


@
Applied General Topology

c© Universidad Politécnica de Valencia

Volume 10, No. 2, 2009

pp. 187-195

Continuous utility functions on submetrizable
hemicompact k-spaces

Alessandro Caterino, Rita Ceppitelli and Francesca Maccarino

Abstract. Some theorems concerning the existence of continuous
utility functions for closed preorders on submetrizable hemicompact k-

spaces are proved. These spaces are precisely the inductive limits of

increasing sequences of metric compact subspaces and in general are

neither metrizable nor locally compact. These results generalize some

well known theorems due to Levin.

2000 AMS Classification: 54F05, 91B16.

Keywords: closed preorder, jointly continuous utility function, hemicom-
pact, submetrizable, k-space.

1. Introduction

The problem concerning the existence of a continuous utility function (pre-
order - preserving map) on a topological space endowed with a (continuous)
preorder is called the utility representation problem. In this paper we study
the utility representation problem in connection with a not necessarily linear
(total) preorder.

Several authors were concerned with non-linear preorders. Indeed, the use
of non-linear preorders (or, more generally, non-linear binary relations) may be
viewed as more realistic and adequate in order to explain the behavior of an
individual. Peleg [27] was the first who presented sufficient conditions for the
existence of a continuous utility function for a partial order on a topological
space. Peleg solved a problem which was posed by Aumann [1] in the context of
expected utility. In particular, Aumann observed that a rational decision-maker
may express “indecisiveness” (or equivalently “incomparability”) between two
alternatives, so that he is not a priori forced to express “indifference” (see also
Ok [26]). Later, Mehta [22, 23] followed the spirit of Nachbin [25] in order to
show that many general results concerning the continuous utility representation
problem can be obtained by combining the classical approach to mathematical



188 A. Caterino, R. Ceppitelli and F. Maccarino

utility theory with some of the most important results in elementary topology.
Herden and Pallack [12] generalized the well known Debreu Theorem concern-
ing the existence of a continuous utility function for a linear preorder on a
second countable topological space. In particular, the authors introduced the
concept of a weakly continuous preorder on a topological space and in this way
provided an appropriate generalization of the notion of a continuous linear pre-
order. Herden and Pallack results imply the interesting Levin Theorem [16]
concerning the existence of a continuous utility function for a not necessarily
linear closed preorder on a second countable locally compact Hausdorff space.
In connection with the above results, we prove a new generalization of Levin
Theorem in the case of closed preorders on spaces not necessarily metrizable
or second countable. More precisely, we prove the existence of a continuous
utility function for a closed preorder on a submetrizable, hemicompact k-space
(submetrizable kω-space).

An interesting survey on hemicompact k-spaces (kω -spaces) can be found in
[11]. Submetrizable kω-spaces are also studied in [7, 8]. We recall that sub-
metrizable kω-spaces are exactly the inductive limits of increasing sequences of
metric compact subspaces [8].

In [6] the authors give some representation theorems assuming topological
conditions that are very close to the ones of our theorems. They prove that if
X is the inclusion inductive limit of a countable chain of compact subspaces
Xn (that is X is a hemicompact k-space) and � is a preorder on X such that
every �|Xn is closed and order-separable, then � is continuously representable.
If we suppose that every �|Xn is a linear preorder without jumps, this result
is equivalent to our result about submetrizable kω -spaces. In general, the two
results are independent.

In the last section we are concerned with the existence of the so-called
jointly continuous utility functions. This kind of problems are often considered
in mathematical economics. Let Γ be a family of preorders (on a topo-
logical space X) endowed with a given topology. The utility representation
problem in this context consists in proving the existence of a continuous map
u : Γ × X −→ R such that u(�, ·) : X −→ R is a utility function for every
�∈ Γ. In mathematical economics X and Γ can be interpreted as a commodity
space and respectively as a space of preference relations of economic agents.

In the second half of the last century, this problem was extensively treated
in the literature. A survey of these results can be found in [5]. We quote
[14, 9, 13, 20, 4] where the existence of jointly continuous functions is proved
in the case of linear preorders. Levin in [16, 17, 18] proved a general theorem
for not necessarily linear preorders in second countable locally compact spaces.
In the present paper we prove a generalization of Levin results in a non-
metrizable setting. More precisely, we prove the existence of a jointly utility
continuous function when X is a submetrizable hemicompact k-space and Γ is
metrizable or both Γ and X are submetrizable hemicompact and the product
Γ × X is a k-space.



Continuous utility functions on submetrizable hemicompact k-spaces 189

2. Preliminaries

A reflexive and transitive binary relation on an arbitrary set X is called a
preorder �. The preorder � on X is linear if [x � y]∨[y � x] for every x, y ∈ X.
An anti-symmetric preorder is said to be an order. If � is a preorder on X the
associated asymmetric relation ≺ is defined by [x ≺ y ⇔ (x � y)∧¬(y � x)]. If
� is a preorder on a set X, then we will refer to the pair (X, �) as a preordered
set.

Let (X, �) be a preordered set. The family of all the sets of the form
]a, +∞[= {x ∈ X : a ≺ x} and ] − ∞, a[= {x ∈ X : x ≺ a}, where a ∈ X,
is a subbasis for a topology on X. This topology, denoted by τ� is the order
topology induced by � and (X, τ�) is called a preordered topological space.
If (X, �) is a preordered set and τ is a topology on X, then the preorder � is
said to be closed with respect to τ if its graph {(x, y) ∈ X × X : x � y} is a
closed subset of X × X. The preorder � is said to be continuous with respect
to τ (or τ -continuous) if for every a ∈ X the sets [a, +∞[= {x ∈ X : a � x}
and ] − ∞, a] = {x ∈ X : x � a} are τ -closed. We recall that if the preorder �
is linear, then � is closed iff it is continuous iff τ� ⊂ τ .
If (X, �) is a preordered set then a real-valued function f on X is said to be

(i) isotone if for every x, y ∈ X, [x � y ⇒ f (x) ≤ f (y)]
(ii) preorder-preserving if f is isotone and [x ≺ y ⇒ f (x) < f (y)].

If (X, �) is a linearly preordered set, then (ii) is equivalent to
(iii) for every x, y ∈ X, [x � y ⇐⇒ f (x) ≤ f (y)].

If the preorder � is interpreted as a preference relation on a set X of alternatives
then a real-valued preorder-preserving function is also called a utility function
or a utility representation of the preorder.

The notions of network and the corresponding netweight seem useful tools
in the theory of representation. Spaces with a countable network are a natural
generalization of the second countable spaces. A network in a topological space
X is a family N of subsets of X such that every open set of X is union of
elements of N . As usual, the net weight of X is defined by

nw(X) = min{|N| : N is a network for X}.

We recall that the netweight is monotone, that is, if X ⊂ Y then nw(X) ≤
nw(Y ).

3. Utility representations for non-linear preorders

We consider the problem of the existence of a continuous utility function
on a topological preordered space when the preorder is not necessarily linear.
We begin with some interesting results due to Levin. We remark that the
techniques used in this case can be very different from those used when � is a
linear preorder. Some results on function spaces are sometimes used to solve
this kind of problems.

If X is a topological space, let C(X, R) be the set of real continuous functions
defined on X. We denote by Cp(X, R) and Ck(X, R) the topological vector



190 A. Caterino, R. Ceppitelli and F. Maccarino

spaces (C(X, R), τp) and (C(X, R), τk), where τp is the topology of pointwise
convergence and τk is the topology of compact convergence.

Theorem 3.1 ([16, Levin]). Let X be a metrizable space such that there exists
a sequence {Xn} of compact subsets of X with the properties:

(1) X =
⋃∞

n=1
Xn;

(2) a real-valued function f on X is continuous if and only if its re-
striction to each Xn is continuous.

Then for every closed preorder � there is a continuous utility function on X.

Note that if X is a second countable locally compact Hausdorff space, then
X satisfies the hypotheses of Theorem 3.1.

Now, the next result extends the above Levin Theorem to non-metrizable
case. We need some definitions and a lemma that is a well known result on
hemicompact k-spaces.

A topological space X is said to be hemicompact if there is a sequence {Kn}
of compact subsets of X which is cofinal in the set of all compact subsets of X,
that is, every compact subset of X is contained in Kn for a suitable n.

A space X is called a k-space if A ⊂ X is open if and only if A ∩ K is open
in K for every compact set K of X, that is, X has the weak topology with
respect to the family of all compact subsets of X.

We recall that the class of the hemicompact k-spaces coincides with the one
of the kω-spaces [11]. A kω -space is a space with a kω-decomposition, that is,
an increasing sequence {Kn} of compact subspaces such that X =

⋃∞
n=1

Kn
and X has the weak topology with respect to the family {Kn}.

Lemma 3.2 ([10]). Let X be a hemicompact k-space and let {Kn} be a sequence
of compact subsets of X which is cofinal in the set of all compact subsets of X.
Then a mapping f : X → Y is continuous if and only if f|Kn is continuous for
every n.

The following lemma generalizes to hemicompact k-spaces an extension the-
orem ([16], Lemma 2) proved by Levin for compact spaces.

Lemma 3.3. Let X be a hemicompact k-space with a closed preorder �, let S
be a compact subset of X and let u : S → [0, 1] be a continuous isotone mapping.
Then there is a continuous isotone function f : X → [0, 1] that extends u.

Proof. Let X =
⋃∞

n=1
Xn where Xn ⊂ X is compact, Xn ⊂ Xn+1 for every

n ∈ N and {Xn} is cofinal in the set of all compact subsets of X. Since S is
compact, S ⊂ Xn̄ for some n̄ ∈ N. By Lemma 2 in [16], the function u can
be extended to Xn̄. Using a recursive process and applying Lemma 3.2, the
function u can be continuously extended to all of X. �



Continuous utility functions on submetrizable hemicompact k-spaces 191

We recall that a topological space X is said to be submetrizable if there
exists a coarser metrizable topology on X.

Theorem 3.4. Let X be a submetrizable space such that there exists a sequence
{Xn} of compact subsets of X with the properties:

(1) X =
⋃∞

n=1
Xn;

(2) a real-valued function f on X is continuous if and only if its re-
striction to each Xn is continuous.

Then for every closed preorder � there is a continuous utility function on X.

Proof. It is not restrictive to suppose that Xn ⊂ Xn+1, for every n ∈ N. As in
the proof of Theorem 2 in [5], if

H = {u ∈ C(X, R) : x � y =⇒ u(x) ≤ u(y)}

is the cone of real continuous isotone mappings defined on X, then x � y if and
only if u(x) ≤ u(y) for every u ∈ H. The hypothesis 1) and the submetrizability
of X imply that X has a countable network. In fact, every compact submetriz-
able Xn is metrizable and second countable. Therefore if Bn = {B

n
i }i∈N is a

countable base of the (metrizable) subspace Xn, then B =
⋃
Bn is a countable

network of X. (C(X, R), τp) has a countable network, too (see [21]). Since the
netweight is monotone, we deduce that also (H, τp) has a countable network
and so it is separable. Let {un}n∈N be a τp-dense sequence in H.
Now, it is possible to show that x � y if and only if un(x) ≤ un(y) for every
n ∈ N. On the contrary, let x, y ∈ X such that un(x) ≤ un(y) for every n ∈ N
but ¬(x � y). Then, using Lemma 3.3, the function u : {x, y} → R defined
by u(y) = 0, u(x) = 1 can be extended to a function ũ ∈ H. Let

Jũ = {g ∈ C(X, R) : |g(x) − ũ(x)| <
1

4
, |g(y) − ũ(y)| <

1

4
}.

Since g(x) − g(y) > 1
2

for every g ∈ Jũ, the density of {un}n∈N in H gets a
contradiction.
Finally, as in Levin’s proof, the function u : X → R defined by

u(x) =

∞∑

n=1

2−n
un(x)

1 + |un(x)|

is the desired continuous utility function.
�

The next result is a direct consequence of Theorem 3.4.

Theorem 3.5. Let X be a submetrizable hemicompact k-space and let � be a
closed preorder on X. Then there is a continuous utility function on X.

We recall that every first countable hemicompact space is locally compact
and second countable (hence metrizable). But, there are submetrizable hemi-
compact k-spaces that are not metrizable spaces.



192 A. Caterino, R. Ceppitelli and F. Maccarino

Example 3.6. Let Z be the set of integers and R be the set of all real num-
bers with the usual topology. Consider the quotient space Y = R/Z obtained
by identifying the set Z to the point y0 ∈ Y . Then Y is a k-space by defi-
nition; it cannot be metrizable since it isn’t even first countable (y0 fails to
have a countable base of neighbourhoods). Moreover, Y is submetrizable and
hemicompact.

Example 3.7. The space S′ of tempered distributions ([7], Example 3.3) is
another example of a submetrizable hemicompact k-space that is not metriz-
able.

In [6] the authors prove some representation theorems assuming topological
conditions that are very close to the assumptions of our Theorems 3.4 and
3.5. In [6], Theorem 2, they prove that if X = lim⊂ Xn (inclusion inductive
limit) where {Xn} is a countable chain of compact subspaces of X and � is a
preorder on X such that every �|Xn is closed and order-separable, then � is
continuously representable.

We note that a space X is the inclusion inductive limit of a countable chain
of compact subspaces if and only if it is a hemicompact k-space. In fact, every
inductive limit of a countable family of compact spaces is a σ-compact k-space
since it is a quotient space of the free union S =

∑
Xn that is σ-compact

and locally compact. The hemicompactness follows by a theorem of Steenrod
in [28]. Conversely, it is easy to prove that every hemicompact k-space is the
inclusion inductive limit of a countable chain of compact subspaces.
If we suppose that �|Xn is a linear preorder without jumps for every n, our
Theorem 3.5 and Theorem 2 in [6] are equivalent. In fact, in this case, since
Xn is compact then τ�|Xn coincides with the topology of the subspace Xn.
Then we get �|Xn is order-separable if and only if Xn is metrizable. Further,
an inclusion inductive limit of a countable chain {Xn} of compact spaces is
submetrizable if and only if Xn is metrizable for every n (see [7, 8]).
In general, our Theorem 3.5 and Theorem 2 in [6] are independent.

4. Jointly continuous utility functions

Let X be a topological space and let Γ be a family of preorders on X,
endowed with a given topology.

The problem of the existence of jointly continuous functions consists in
proving the existence of a continuous map u : Γ × X −→ R such that
u(�, ·) : X −→ R is a utility function for every �∈ Γ.

In [16] Levin, using a continuous selection theorem of Michael [24], solved
the problem when Γ is metrizable and X is locally compact and second
countable. In the special case in which Γ and X are both locally compact
and second countable, Levin proved the result without the use of Michael’s
Theorem.

We begin to extend this result to the case when Γ and X are submetrizable
and hemicompact and Γ×X is a k-space. Later (Theorem 4.2) we will extend
the Theorem of Levin to the case when X is submetrizable and hemicompact,



Continuous utility functions on submetrizable hemicompact k-spaces 193

Γ is metrizable and Γ × X is a k-space. We note that these two results are
independent of each other.

Theorem 4.1. Let Γ and X be submetrizable hemicompact spaces and sup-
pose Γ × X is a k-space. Moreover, assume that the set

G = {(�, x, y) : x � y}

is closed in Γ × X × X.
Then there exists a continuous function u : Γ × X → [0, 1] such that, for each
� ∈ Γ, u(� , ·) is a continuous utility function on X.

Proof. As in Section 5 of [16], the preorder ≤ on Γ × X, defined by
(�1 , x1) ≤ (�2 , x2) if and only if �1 = �2 and x1 �1 x2, is closed.

Hence (Γ × X, ≤ ) satisfies all the conditions of Theorem 3.5 and so there is
a continuous utility function u on Γ × X with respect to ≤ . The conclusion
follows from the definition of the preorder ≤.

�

Theorem 4.2. Let Γ be a metrizable space, let X be a submetrizable hemi-
compact space and suppose Γ × X is a k-space. Moreover, assume that the
set

G = {(�, x, y) : x � y}

is closed in Γ × X × X.
Then there exists a continuous function u : Γ × X → [0, 1] such that, for each
� ∈ Γ, u(� , ·) is a continuous utility function on X.

Proof. Let {Kn} be a sequence of compact subsets of X which is cofinal in the
set of all compact subsets of X. Then the topology τk on Ck(X, R) is generated
by the countable family of seminorms pn(f ) = supx∈Kn | f (x) | . Therefore,
the locally convex topological vector space Ck(X, R) is metrizable. Moreover,
since X is a k-space then Ck(X, R) is also complete ([15], Theorem 12) and
hence it is a Fréchet space. As in the proof of Theorem 3.4, the hypotheses of
hemicompactness and submetrizability imply that X has a countable network
and so Cp(X, R) is separable. Then Ck(X, R) is separable too ([21], Cor. 4.2.2).
Finally, by the hypotheses that Γ × X is a k-space, using the same arguments
as in Theorem 1 in [16], it is possible to construct the desired jointly continuous
utility function u.

�

Acknowledgements. The authors would like to thank Prof. G. Bosi for
his appreciated suggestions and comments.



194 A. Caterino, R. Ceppitelli and F. Maccarino

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Received November 2008

Accepted November 2009

Alessandro Caterino (caterino@dmi.unipg.it)
Dipartimento di Matematica e Informatica, Università degli Studi di Perugia,
Via Vanvitelli 1, 06123, Perugia, Italy (corresponding author)

Rita Ceppitelli (matagria@dmi.unipg.it)
Dipartimento di Matematica e Informatica, Università degli Studi di Perugia,
Via Vanvitelli 1, 06123, Perugia, Italy

Francesca Maccarino (maccarinofrancesca@libero.it)
Via Pian del Vantaggio 10A, 05018 Orvieto (Terni), Italy