Continuous semiorder representations (Q1099048)

Given \(<X,\succ >\), where X is a topologically connected space and \(\succ\) is an asymmetric binary relation, necessary and sufficient conditions are presented for the existence of a continuous representation of the form \(<u,\delta >\); that is, for x,y\(\in X\), \(x\succ y\) if and only if \(u(x)>u(y)+\delta\) where \(u: X\to {\mathbb{R}}\) is continuous and \(\delta\) is a strictly positive real number. The results are related to existing results for numerical representations of interval orders and semiorders.