The algebraic versus the topological approach to additive representations (Q1115377)

Let \({\mathcal C}_ 1,...,{\mathcal C}_ n\) be connected topological spaces. Suppose that each of these spaces represents the range set for an attribute that some object may possess. Then a point in the Cartesian product \(\prod^{n}_{i=1}{\mathcal C}_ i\) can be regarded as a collection of attribute values. One concern of measurement theory is understanding how one can endow such attribute collections with a preference order. One way in which an order can be established is by the use of a function \(V: \prod {\mathcal C}_ i\to {\mathbb{R}}.\) If V is obtained as the sum of functions \(V_ i(x_ i)\) it is called additive. It is of interest to know when such an additive function is continuous. A coordinate is inessential for a preference function if replacing the value of that coordinate by another value yields an element of equivalent preference. A coordinate is essential if it is not inessential. The main result of this paper is the following. Let V: \(\prod {\mathcal C}_ i\to {\mathbb{R}}\). Let the following five conditions hold. (i) V is additive; (ii) \({\mathcal C}_ 1,...,{\mathcal C}_ n\) are connected topological spaces; (iii) \(\prod {\mathcal C}_ i\) has the product topology; (iv) The binary relation \(\succcurlyeq\) on \(\prod {\mathcal C}_ i\), represented by V, is continuous; (v) At least two coordinates are essential. Then this additive function is continuous.