In this paper, we provide a characterization of the existence of a representation of an interval order on a topological space in the general case by means of a pair of continuous functions, when neither the functions nor the topological space are required to satisfy any particular assumptions. Such a characterization is based on a suitable continuity assumption of the binary relation, called weak continuity . In this way, we generalize all the previous results on the continuous representability of interval orders, and also of total preorders, as particular cases.

中文翻译：

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用两个连续函数连续表示区间阶数

在本文中，我们在一般情况下通过一对连续函数提供了拓扑空间上区间阶表示的存在性的表征，此时函数和拓扑空间都不需要满足任何特定假设. 这种表征基于二元关系的适当连续性假设，称为弱连续性。通过这种方式，我们将所有先前的结果概括为区间阶数的连续可表示性，以及总预阶数的连续可表示性，作为特殊情况。