  Let $X$ be an arbitrary set. Then a topology $t$ on $X$ is said to be
completely useful if every upper semicontinuous linear (total) preorder
$\precsim$ on $X$ can be represented by an upper semicontinuous real-valued
order preserving function. In this paper, appealing, simple and new
characterizations of completely useful topologies will be proved, therefore
clarifying the structure of such topologies.
