<blockquote>
<p>Theorem (<a href="https://cowles.yale.edu/sites/default/files/files/pub/d00/d0076.pdf" rel="nofollow noreferrer">Debreu 1959</a> page 9, 10) Let <span class="math-container">$X$</span> be connected separable topological space endowed with product topology. <em>If <span class="math-container">$\succsim$</span> is independent and at least three factors are essential, then there exist functions <span class="math-container">$u_i:X_i\rightarrow \mathbb{R}$</span> such that</em>
<span class="math-container">$$
x\succsim y\iff \sum_{i=1}^n u_i (x_i) \geqslant \sum_{i=1}^n u_i (y_i).
$$</span></p>
</blockquote>
<p><strong>Question</strong>: can the assumption of <strong>&quot;connected separable</strong>&quot; be changed to &quot;<strong>second countable</strong>&quot;?</p>
<p>I think the primary reason of using &quot;connected separable&quot; is for solvability and cardinality restriction, and second countability is usually stronger than topological separability.</p>
<p><strong>Motivation:</strong> Debreu' representation theorem of <em>ordinal</em> utility uses either &quot;connected separable&quot; or &quot;second countable&quot;. Many follow-up of Debreu's work in ordinal utility uses the assumption of &quot;second countable&quot;, not &quot;connected separable&quot;. This leads to a diverge of assumptions in two strands of literature.</p>

CC BY-SA 4.0
2022-05-26T11:13:41.067Z
Debreu's theorem is stronger; the additive representation is unique up to a positive linear representation. This uniqueness will certainly fail without connectedness; you can construct counterexamples with $X$ finite.
Michael Greinecker
CC BY-SA 4.0
2022-05-26T19:54:20.557Z
@MichaelGreinecker Good point!
High GPA
CC BY-SA 4.0
2023-09-14T12:10:15.463Z
@MichaelGreinecker If I am only interested in the existence of utility function (not cardinal uniqueness), can the assumption of connected separable be relaxed to second countable?
High GPA
CC BY-SA 4.0
2023-09-14T13:15:16.657Z
I don't know the answer.
Michael Greinecker
<microeconomics><utility><decision-theory><theory>
Can the topological assumption in Debreu's representation theorem of cardinal utility be altered from "connected separable" to "second countable"?
High GPA
# Can the topological assumption in Debreu's representation theorem of cardinal utility be altered from "connected separable" to "second countable"?

(Asked by: username_0 on 2022-05-25)

> Theorem ([Debreu 1959](https://cowles.yale.edu/sites/default/files/files/pub/d00/d0076.pdf) page 9, 10) Let $X$ be connected separable topological space endowed with product topology. *If $\succsim$ is independent and at least three factors are essential, then there exist functions $u\_i:X\_i\rightarrow \mathbb{R}$ such that*
> $$
> x\succsim y\iff \sum\_{i=1}^n u\_i (x\_i) \geqslant \sum\_{i=1}^n u\_i (y\_i).
> $$

**Question**: can the assumption of **"connected separable**" be changed to "**second countable**"?

I think the primary reason of using "connected separable" is for solvability and cardinality restriction, and second countability is usually stronger than topological separability.

**Motivation:** Debreu' representation theorem of *ordinal* utility uses either "connected separable" or "second countable". Many follow-up of Debreu's work in ordinal utility uses the assumption of "second countable", not "connected separable". This leads to a diverge of assumptions in two strands of literature.

**Tags:** microeconomics, utility, decision-theory, theory
> **Comments:**
> **username_1**: Debreu's theorem is stronger; the additive representation is unique up to a positive linear representation. This uniqueness will certainly fail without connectedness; you can construct counterexamples with $X$ finite.  
> **username_0**: @MichaelGreinecker Good point!  
> **username_0**: @MichaelGreinecker If I am only interested in the existence of utility function (not cardinal uniqueness), can the assumption of connected separable be relaxed to second countable?  
> **username_1**: I don't know the answer.  

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## 0 answers

