^{1}

^{*}

^{2}

^{*}

We present different conditions for the existence of a pair of upper semicontinuous functions representing an interval order on a topological space without imposing any restrictive assumptions neither on the topological space nor on the representing functions. The particular case of second countable topological spaces, which is particularly interesting and frequent in economics, is carefully considered. Some final considerations concerning semiorders finish the paper.

An interval order

The existence of numerical representations of interval orders was first studied by Fishburn [

When the set X is endowed with a topology

For many purposes, the existence of a representation

In this paper, we present different results concerning the representability of an interval order

An interval order

An interval order

It is well know that if

Fishburn [

The following proposition was proved, for example, by Alcantud et al. ([

Proposition 2.1. Let

An interval order

Therefore, we have that

An interval order

From Chateauneuf [

If R is a binary relation on a set X, then denote by

A subset A of a related set

A real-valued function u on X is said to be a weak utility function for a partial order

The following characterization of the existence of an upper semicontinuous weak utility for a partial order on a topological space is well known (see e.g. Alcantud and Rodríguez-Palmero ([

Proposition 2.2. Let

1) There exists an upper semicontinuous weak utility function u for

2) There exists a countable family

A real-valued function u on X is said to be a utility function for a partial order

If a partial order

The following proposition is well known and easy to be proved.

Proposition 2.3. Let

1) There exists an upper semicontinuous utility function u for

2) There exists a countable family

A pair

If

We say that a pair

An interval order

is an open subset of X for every

If there exists a representation

is open for every

is expressed as union of open sets.

On the other hand, the existence of an upper semicontinuous representation does not imply that the weak order

Example 2.4. Let X be the set

If we define

A weak order

If

1)

2)

The following proposition is a particular case of Theorem 4.1 in Burgess and Fitzpatrick [

Proposition 2.5. If

defines an upper semicontinuous function on

In the following theorem we present some conditions that are equivalent to the existence of an upper semicontinuous representation of an interval order on a topological space.

Theorem 3.1. Let

1) There exists a pair

2) The following conditions are verified:

a) The interval order

b)

c) There exists an upper semicontinuous weak utility

3) The following conditions are verified:

a) The interval order

b)

c) There exists a countable family

4) There exists a countable family

5) There exists a countable family

a)

b)

c) for all

6) There exist two quasi scales

a)

b) for every

Proof. The equivalence 1) Û 5) was proved in Bosi and Zuanon ([

Let us prove the equivalence of conditions 1) and 4). It is clear that 1) implies 4). In order to show that 4) implies 1), assume that there exists a countable family

in order to immediately verify that

Finally, let us show that also the equivalence of conditions 1) and 6) is valid. In order to show that 1) implies 6), assume without loss of generality that there exists a pair of upper semicontinuous real-valued functions with values in

verify that

fies the above subconditions a) and b) of condition 6).

In order to show that 6) implies 1), assume that there exist two quasi scales

the family

as follows:

We claim that

From the definition of the functions u and v, it is clear that they both take values in

Finally, observe that u and v are upper semicontinuous real-valued functions on

It has been noticed that if

Corollary 3.2. Let

1) There exists a pair

2) The following conditions are verified:

a) The interval order

b)

c)

3) The following conditions are verified:

a) The interval order

b)

c) There exists a countable family

Since Bridges ([

Corollary 3.3. Let

1) There exists a pair

2) The following conditions are verified:

a)

b) There exists an upper semicontinuous weak utility

The following corollary is a consequence of both Corollary 3.2 and Corollary 3.3.

Corollary 3.4. Let

1) There exists a pair

2) The following conditions are verified:

a)

b)

The following corollary is found in Bosi and Zuanon ([

Corollary 3.5. Let

1) There exists a pair

2) The following conditions are verified:

a)

b)

We finish this paper by presenting some applications of the previous results to the semiorder case. We recall that a semiorder

If

Proposition 3.6. Let

Clearly, this happens in the particular case when

Proposition 3.7. Let

Since it was already observed that upper semicontinuity of an interval order

Corollary 3.8. Let

Corollary 3.9. Let