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EnglishWe discuss the continuous real representability of a not necessarily total preorder on a normal topological space in connection with a suitable continuity assumption, called C-continuity in this paper. We show that a topology $\tau$ on a set $X$ is normal if and only if the topological preordered space $(X, \precsim , \tau )$ is normally preordered for every $C$-continuous preorder $\precsim$ on $(X, \tau )$. We also prove that a C-continuous preorder $\precsim$ on a normal topological space $(X, \tau )$ is representable by means of a continuous order-preserving function $u$ if and only if $\precsim$ verifies a suitable separability condition \`a la Nachbin.
| Titolo: | Continuous order-preserving functions for nontotal preorders on normal spaces |
| Autori: | |
| Data di pubblicazione: | 2012 |
| Rivista: | |
| Abstract: | We discuss the continuous real representability of a not necessarily total preorder on a normal topological space in connection with a suitable continuity assumption, called C-continuity in this paper. We show that a topology $\tau$ on a set $X$ is normal if and only if the topological preordered space $(X, \precsim , \tau )$ is normally preordered for every $C$-continuous preorder $\precsim$ on $(X, \tau )$. We also prove that a C-continuous preorder $\precsim$ on a normal topological space $(X, \tau )$ is representable by means of a continuous order-preserving function $u$ if and only if $\precsim$ verifies a suitable separability condition \`a la Nachbin. |
| Handle: | http://hdl.handle.net/11368/2641871 |
| Appare nelle tipologie: | 1.1 Articolo in Rivista |
