The descriptive set-theoretic complexity of the set of points of continuity of a multi-valued function
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_\del...
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| Format: | Article |
| Language: | English |
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Logical Methods in Computer Science e.V.
2011-11-01
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| Series: | Logical Methods in Computer Science |
| Subjects: | |
| Online Access: | https://lmcs.episciences.org/887/pdf |
| _version_ | 1797988573501194240 |
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| author | Vassilios Gregoriades |
| author_facet | Vassilios Gregoriades |
| author_sort | Vassilios Gregoriades |
| collection | DOAJ |
| description | 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}]. |
| first_indexed | 2024-04-11T08:06:17Z |
| format | Article |
| id | doaj.art-a7e239b0f3df4c6ea1f78736e5cfbf52 |
| institution | Directory Open Access Journal |
| issn | 1860-5974 |
| language | English |
| last_indexed | 2024-04-11T08:06:17Z |
| publishDate | 2011-11-01 |
| publisher | Logical Methods in Computer Science e.V. |
| record_format | Article |
| series | Logical Methods in Computer Science |
| spelling | doaj.art-a7e239b0f3df4c6ea1f78736e5cfbf522022-12-22T04:35:34ZengLogical Methods in Computer Science e.V.Logical Methods in Computer Science1860-59742011-11-01Volume 7, Issue 410.2168/LMCS-7(4:2)2011887The descriptive set-theoretic complexity of the set of points of continuity of a multi-valued functionVassilios GregoriadesIn 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}].https://lmcs.episciences.org/887/pdfmathematics - logiccomputer science - logic in computer sciencef.4.1 |
| spellingShingle | Vassilios Gregoriades The descriptive set-theoretic complexity of the set of points of continuity of a multi-valued function Logical Methods in Computer Science mathematics - logic computer science - logic in computer science f.4.1 |
| title | The descriptive set-theoretic complexity of the set of points of continuity of a multi-valued function |
| title_full | The descriptive set-theoretic complexity of the set of points of continuity of a multi-valued function |
| title_fullStr | The descriptive set-theoretic complexity of the set of points of continuity of a multi-valued function |
| title_full_unstemmed | The descriptive set-theoretic complexity of the set of points of continuity of a multi-valued function |
| title_short | The descriptive set-theoretic complexity of the set of points of continuity of a multi-valued function |
| title_sort | descriptive set theoretic complexity of the set of points of continuity of a multi valued function |
| topic | mathematics - logic computer science - logic in computer science f.4.1 |
| url | https://lmcs.episciences.org/887/pdf |
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