Abstract Continuous functions over compact Hausdorff spaces have been completely characterised. We consider the more general problem: given a set-valued function T from an arbitrary set X to itself, does there exist a compact Hausdorff topology on X with respect to which T is upper semicontinuous? We give conditions that are necessary for T to be upper semicontinuous and point-closed if X is a compact Hausdorff space. We show that it is always possible to provide X with a compact T1 topology with respect to which T is lower semicontinuous, and consequently, if T : X → X is a function, then it is always possible to provide X with a compact T1 topology with respect to which T is continuous.

中文翻译：

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涉及集值函数的抽象拓扑动力学

摘要 紧致 Hausdorff 空间上的连续函数已被完全表征。我们考虑更一般的问题：给定一个从任意集合 X 到其自身的集合值函数 T，是否存在 X 上关于 T 是上半连续的紧凑豪斯多夫拓扑？如果 X 是紧致 Hausdorff 空间，我们给出 T 为上半连续和点闭的必要条件。我们表明总是可以为 X 提供一个紧凑的 T1 拓扑，而 T 是下半连续的，因此，如果 T : X → X 是一个函数，那么总是可以为 X 提供一个紧凑的 T1 拓扑关于 T 是连续的。