Order embedding theorems and multi-utility representation of the preorder (Q6592024)

Given a preordered topological space \(( X, \tau, \leq)\), let \(C_\uparrow(X,[0,1])\) be the set of continuous increasing functions from \(X\) to \([0,1]\). The space \((X, \tau, \leq)\) is completely regular ordered if (a) \(x \not\leq y\) implies there exists \(f \in C_\uparrow(X,[0,1])\) with \(f(y) <f(x)\) and (b) if \(F\) is a closed set of \(X\) and \(x \not\in F\), then there exist \(f, g \in C_\uparrow(X,[0,1])\) with \(1-f(x) = g(x) = 1\) and \((1-f(y)) \wedge g(y) = 0\) for all \(y \in F\). Following the construction of the Stone-Čech compactification of a completely regular topological space, \textit{L. Nachbin} [Topology and order. Translated from the Portuguese by Lulu Bechtolsheim. Princeton-New-Jersey-Toronto-New York-London: D. Van Nostrand Company, Inc. (1965; Zbl 0131.37903)] showed that a completely regular (pre)ordered topological spaces is homeomorphic and order isomorphic to a subspace of the Tychonoff cube \([0,1]^{C_\uparrow(X,[0,1])}\). This is a very large cube. If \(w(X)\) is the weight of \(X\), the author shows that the completely regular (pre)ordered topological space \((X, \tau, \leq)\) can be embedded in a cube \([0,1]^\mathfrak{m}\) where \(\mathfrak{m} = w(X) + \aleph_0\). Among other results which generalize the theory of compactifications and ordered compactifications, the authors show that, given a compact subspace \(K\) of a completely regular preordered topological space \((X, \tau, \leq)\), there is an embedding of \(X\) into \([0,1]^\mathfrak{m}\) such that \(e(K)\) is contained in a product \(\prod \{ \Delta_\alpha : \alpha < w(K) + \aleph_0\}\), where each \(\Delta_\alpha\) is a diagonal of some Tychonoff cube \([0,1]^{\kappa_\alpha}\) and \(\prod \{ [0,1]^{\kappa_\alpha} : \alpha < w(K) + \aleph_0\} = [0,1]^{w(X) + \aleph_0}\). The author nicely describes how the results relate to the study of utility theory in economics.