The Euclidean distance construction of order homomorphisms

The Euclidean distance technique of Arrow and Hahn is used to construct an upper semicontinuous order homomorphism (partial utility function) from (X, ≻) to ($\text{R}$, >), where X is a closed, convex subset of $\text{R}$^N and ≻ is a continuous strict partial order on X. It is also shown that the order homomorphism is upper semicontinuous as a function on $\text{X×}\text{P}\text{(X)}$, where$\text{P}\text{(X)}$ is the set of continuous strict partial orders on X, taken with the topology of closed convergence.