Continuous Order‑Preserving Functions for All Kind of Preorders

A topology is said to be strongly useful if every weakly continuous preorder admits a continuous order-preserving function. A strongly useful topology is useful, in the sense that every continuous total preorder admits a continuous utility representation. In this paper, I study the structure of strongly useful topologies. The existence of a natural one-to-one correspondence is proved, between weakly continuous preorders and equivalence classes of families of complete separable systems. In some sense, this result completely clarifies the connections between order theory and topology. Then, I characterize strongly useful topologies and I present a property concerning subspace topologies of strongly useful topological spaces.