The structure of useful topologies

Let $X$ be an arbitrary set. A topology $t$ on $X$ is said to be useful if every complete and continuous preorder on $X$ is representable by a continuous real-valued order preserving function. It will be shown, in a first step, that there exists a natural one-to-one correspondence between continuous and complete preorders and complete separable systems on $X$. This result allows us to present a simple characterization of useful topologies $t$ on $X$. According to such a characterization, a topology $t$ on $X$ is useful if and only if for every complete separable separable system $cal E$ on $(X,t)$ the topology $t_{cal E}$ generated by $cal E$ and by all the sets $X setminusoverline{E}$ is second countable. Finally, we provide a simple proof of the fact that the countable weak separability condition ({em cwsc}), which is closely related to the countable chain condition ({em ccc}), is necessary for the usefulness of a topology.