The descriptive set-theoretic complexity of the set of points of continuity of a multi-valued function
The descriptive set-theoretic complexity of the set of points of continuity of a multi-valued function
复制标题
多值函数连续点集的描述性集合论复杂性
DOI：
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发表时间：
2011
期刊：
影响因子：
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通讯作者：
Vassilios Gregoriades
中科院分区：
文献类型：
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作者：
Vassilios Gregoriades
在这篇文章中，我们处理一个概念的连续性为一个多值函数$F$和我们计算的描述集理论复杂性的集合的所有$x$，其中$F$是连续的$x$。我们给条件下，后者的集合是一个$G_delta$集或可数联盟的$G_delta$集。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$F$的连续点集是Borel，即，我们表明，如果我们放弃一些以前的条件，然后有一个多值函数F$的图形是一个博雷尔集和一组点的连续性F$是不是一个博雷尔集。最后，我们给出了一些类似的结果，一个更强的概念，连续性的多值函数。这篇文章是由M.齐格勒在[{em真实的计算与最小离散建议：非均匀可计算性的复杂性理论与应用线性代数}，{sl提交}]。
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 $G_delta$ set or the countable union of $G_delta$ sets. 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 analogous results regarding a stronger notion of continuity for a multi-valued function. This article is motivated by a question of M. Ziegler in [{em Real Computation with Least Discrete Advice: A Complexity Theory of Nonuniform Computability with Applications to Linear Algebra}, {sl submitted}].
具有最小离散建议的实际计算：非均匀可计算性的复杂性理论及其在有效线性代数中的应用
DOI：
10.1016/j.apal.2011.12.030
发表时间：
2012
期刊：
ArXiv
影响因子：
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作者：
M. Ziegler
通讯作者：
M. Ziegler