The structure of useful topologies

Abstract

Abstract 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 system E on ( X , t ) the topology t E generated by E and by all the sets X ∖ E ¯ is second countable. Finally, we provide a simple proof of the fact that the countable weak separability condition (cwsc), which is closely related to the countable chain condition (ccc), is necessary for the usefulness of a topology.