The structure of useful topologies

Gianni Bosi and Gerhard Herden

Journal of Mathematical Economics, 2019, vol. 82, issue C, 69-73

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 tE 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.

Keywords: Complete preorder; Complete separable system; Continuity; Countability; Locally finiteness (search for similar items in EconPapers)
Date: 2019
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Citations: View citations in EconPapers (5)

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Persistent link: https://EconPapers.repec.org/RePEc:eee:mateco:v:82:y:2019:i:c:p:69-73

DOI: 10.1016/j.jmateco.2019.02.006

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