<p><a href="https://en.wikipedia.org/wiki/Debreu%27s_representation_theorems" rel="nofollow noreferrer">Debreu's representation theorems</a></p>
<p>Debreu 1959 states that: second countability, continuity, and weak ordering sufficiently implies the existence of real (continuous) utility function. The second and third assumptions are also necessary.</p>
<p>My question is, can we prove that the second countability is not necessary by providing a counterexample?</p>

CC BY-SA 4.0
2023-09-14T13:20:24.800Z
None of the conditions besides weak ordering are necessary. A constant function on a topological space that fails to be second countable gives the desired counterexample.
Michael Greinecker
CC BY-SA 4.0
2023-09-14T14:31:58.187Z
@MichaelGreinecker I disagree. A constant utility means all points are equivalent, which induces a second countable topology!
High GPA
CC BY-SA 4.0
2023-09-14T14:47:02.420Z
Trivial topology is second countable!
High GPA
CC BY-SA 4.0
2023-09-14T16:02:34.000Z
Yes. But the underlying space need not be trivial. The topology generated by a utility function is necessarily second countable, but that is a different issue.
Michael Greinecker
CC BY-SA 4.0
2023-10-03T18:38:43.143Z
@MichaelGreinecker You are right. My question needs to be more precise by adding some statements like "induced topology also counts"
High GPA
CC BY-SA 4.0
2023-10-03T21:22:56.350Z
Btw: The Wikipedia article is riddled with mistakes.
Michael Greinecker
CC BY-SA 4.0
2023-10-04T07:33:12.323Z
@MichaelGreinecker I agree. I'll find a better source
High GPA
<microeconomics><utility><theory><decision-theory><choice-theory>
In Debreu's representation theorem of ordinal utility, is the assumption of "second countability" necessary?
High GPA
# In Debreu's representation theorem of ordinal utility, is the assumption of "second countability" necessary?

(Asked by: username_0 on 2023-09-14)

[Debreu's representation theorems](https://en.wikipedia.org/wiki/Debreu%27s_representation_theorems)

Debreu 1959 states that: second countability, continuity, and weak ordering sufficiently implies the existence of real (continuous) utility function. The second and third assumptions are also necessary.

My question is, can we prove that the second countability is not necessary by providing a counterexample?

**Tags:** microeconomics, utility, theory, decision-theory, choice-theory

> **Comments:**
> **username_1**: None of the conditions besides weak ordering are necessary. A constant function on a topological space that fails to be second countable gives the desired counterexample.  
> **username_0**: @username_1 I disagree. A constant utility means all points are equivalent, which induces a second countable topology!  
> **username_0**: Trivial topology is second countable!  
> **username_1**: Yes. But the underlying space need not be trivial. The topology generated by a utility function is necessarily second countable, but that is a different issue.  
> **username_0**: @username_1 You are right. My question needs to be more precise by adding some statements like "induced topology also counts"  
> **username_1**: Btw: The Wikipedia article is riddled with mistakes.  
> **username_0**: @username_1 I agree. I'll find a better source  

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## 0 answers

