In this paper, we present a new simple axiomatization of useful topologies , i.e., topologies on an arbitrary set, with respect to which every continuous total preorder admits a continuous utility representation. In particular, we show that, for completely regular spaces, a topology is useful, if and only if it is separable, and every isolated chain of open and closed sets is countable. As a specific application to optimization theory, we characterize the continuous representability of all continuous total preorders, which admit both a maximal and a minimal element.

中文翻译：

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所有连续总预购的连续可表示性的拓扑

在本文中，我们提出了一种新的有用拓扑的简单公理化，即任意集合上的拓扑，关于它，每个连续的总前序都允许连续的效用表示。特别是，我们表明，对于完全规则的空间，拓扑是有用的，当且仅当它是可分的，并且每个孤立的开集和闭集链都是可数的。作为优化理论的一个具体应用，我们描述了所有连续总前序的连续可表示性，它允许一个最大和一个最小元素。