Semicontinuous utility functions in topological spaces

Abstract

Rader [7] in 1963 gave a proof of a theorem about the existence of an upper semicontinuous function on a second countable topological spaceX with a total preorder\(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \prec } \), provided that the left half lines\(\left\{ {x : x \prec y} \right\}\) are open for eachy belonging toX. The proof of Rader was constructive and direct, without any use of theorems like the Continuoum Characterization Theorem.