多值函数连续点集的描述性集合论复杂性
The descriptive set-theoretic complexity of the set of points of continuity of a multi-valued function (Extended Abstract)
摘要：
在这篇文章中，我们处理一个多值函数F的连续性的概念，我们计算的集合的描述集理论的复杂性的所有x的F是连续的在x。我们给出了后者是G_δ集或G_δ集的可数并的条件。并给出了一个反例，说明在相同条件下，后一个结果是最优的.此外，我们证明了这些条件是必要的，以获得点的连续F的集合是Borel，即，我们表明，如果我们放弃一些以前的条件，那么有一个多值函数F的图是一个Borel集和F的连续点的集合是不是一个Borel集。最后，我们给出了一些类似的结果，一个更强的概念的连续性的多值函数。这篇文章是由M. Ziegler在“Real Computation with Least Discrete Advice：A Complexity Theory of Nonuniform Computability with Applications to Linear Algebra”中的文章。
英文摘要：
In this article we treat a notion of continuity for a multi-valued function F and we compute the descriptive set-theoretic complexity of the set of all x for which F is continuous at x. We give conditions under which the latter set is either a G_δset or the countable union of G_δsets. Also we provide a counterexample which shows that the latter result is optimum under the same conditions. Moreover we prove that those conditions are necessary in order to obtain that the set of points of continuity of F is Borel i.e., we show that if we drop some of the previous conditions then there is a multi-valued function F whose graph is a Borel set and the set of points of continuity of F is not a Borel set. Finally we give some analogue results regarding a stronger notion of continuity for a multi-valued function. This article is motivated by a question of M. Ziegler in "Real Computation with Least Discrete Advice: A Complexity Theory of Nonuniform Computability with Applications to Linear Algebra", (submitted).
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