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.

### The structure of useful topologies

#### 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 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.
##### Scheda breve Scheda completa
2019
26-feb-2019
Pubblicato
https://www.sciencedirect.com/science/article/abs/pii/S0304406819300254
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11368/2939404`
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