# Does Debreu's representation theorem of ordinal utility require Hausdorff topology?

By Debreu's theorem of ordinal utility, any continuous weak order on $$X$$ is represented with a continuous utility function, if $$X$$ is a second countable or connected separable topological space.

My question is, does the theorem require $$X$$ to be endowed with T0 or T1 or T2 (Hausdorff) topology?

## 1 Answer

No. However, the problem can be reduced to representing preferences on a Hausdorff space. Instead of trying to represent a complete preorder on a set, one can try to represent linear orders on the space of indifference classes. On the latter space, Hausdorffness is automatic if the preference relation is continuous. To see this, let $$X$$ be a topological space and $$\succeq$$ be a complete preorder on $$X$$ such that for every $$x\in X$$, the sets $$\{y\in X\mid y\succeq x\}$$ and $$\{y\in X\mid x\succeq y\}$$ are closed. Define $$\succ$$, $$\sim$$ in the usual way and endow $$X/\sim$$ with the quotient topology. Then $$X/\sim$$ is a Hausdorff space. Write $$[x]$$ for the equivalence class corresponding to $$x$$ and view $$\succeq$$ by abuse of notation as a linear order on $$X/\sim$$. Now, let $$[x]\neq [y]$$. Without loss of generality, let $$x\prec y$$.

There are two cases: Either there exists $$z\in X$$ such that $$x\prec z\prec y$$ or there does not. If there exists such a $$z$$, then the sets $$\{v\mid v\prec z\}$$ and $$\{v\mid v\succ z\}$$ are open neighborhoods of $$x$$ and $$y$$, respectively. By the definition of the quotient topology, the sets $$\{[v]\mid v\prec z\}$$ and $$\{[v]\mid v\succ z\}$$ are open neighborhoods of $$[x]$$ and $$[y]$$, respectively. If there exists no such $$z$$, then the sets $$\{v\mid v\prec y\}$$ and $$\{v\mid v\succ x\}$$ are open neighborhoods of $$x$$ and $$y$$, respectively. Again by the definition of the quotient topology, the sets $$\{[v]\mid v\prec y\}$$ and $$\{[v]\mid v\succ x\}$$ are open neighborhoods of $$[x]$$ and $$[y]$$, respectively.

It follows that $$X/\sim$$ is a Hausdorff space and the problem can be reduced to representing a preference ordering on a Hausdorff space.

• Wow where did you learn all those things? So impressive. May 14 at 16:35
• Thank you; one can learn a lot over time. May 14 at 17:21