AI 中文总结
本文在序理论与拓扑框架下，研究无限备选集合上的二元关系，给出满足上麦克尼尔信息单调性的一致抽象决策问题中，vNM稳定极大性的拓扑刻画，拓展了最优选择集理论的应用场景。
AI 中文摘要
最优选择集理论是社会选择与博弈论中成熟的框架。当偏好呈现循环性时，这在复杂经济环境中十分常见，极大元集合可能为空，因此催生了冯·诺依曼-摩根斯坦（vNM）稳定集等替代解概念。本文在序理论与拓扑框架下研究无限备选集合上的二元关系，主要结果是给出了冯·诺依曼-摩根斯坦（vNM）稳定极大性的拓扑刻画：对于满足上麦克尼尔信息单调性的一致抽象决策问题，极大元集合非空且稳定，当且仅当存在集合X上的紧拓扑，使得关系R是纳赫宾闭的且上半连续的。
英文摘要
The theory of optimal choice sets provides a well-established framework in social choice and game theory. When preferences are cyclic, as often occurs in complex economic environments, the set of maximal elements may be empty, thereby motivating alternative solution concepts such as the von Neumann--Morgenstern (vNM) stable set. In this paper, we study binary relations on infinite sets of alternatives within an order-theoretic and topological framework. Our main result yields a topological characterization of von Neumann--Morgenstern stable maximality: for consistent abstract decision problems satisfying Upper MacNeille Informational Monotonicity, the set of maximal elements is non-empty and stable if and only if there exists a compact topology on \(X\) with respect to which \(R\) is Nachbin closed and upper semicontinuous.
