×

Concerning continuity apart from a meager set. (English) Zbl 0606.54008

This paper is motivated by the well-known theorem of K. Kuratowski [Fundam. Math. 16, 390-394 (1930)] which says that if I is a \(\sigma\)- ideal on a space X and Y is a space with a countable base, then a mapping f from X into Y is continuous apart from an element of I iff for every open set \(U\subset Y\) there exist an open set \(V\subset X\) and A,B\(\in I\) such that \(f^{-1}(U)=(U-A)\cup B\). The mapping f is called I-continuous if for every open set \(U\subset Y\) there exist an open set \(V\subset X\) and \(A\in I\) such that \(f^{-1}(U)=V-A\). The authors prove the following theorem: if I is a \(\sigma\)-ideal on a space X, Y is a regular space, f is a mapping from X into Y and either Y has a countable base or X is hereditarily Lindelöf or I consists of all meager sets of X, then f is I-continuous iff there exists a closed set \(F\subset X\) such that \(f| F\) is continuous and X-F\(\in I\). As an application of this theorem they obtain a very strong improvement of the theorem of W. Sierpiński and A. Zygmund [ibid. 4, 316-318 (1923)] that there exists a function \(f: {\mathbb{R}}\to {\mathbb{R}}\) such that \(f| X\) is discontinuous for every \(X\in {\mathbb{R}}\) of power continuum. Namely they prove that if both X and Y are metric, X is separable and Y is complete and all subsets of X of power less than \(2^{\omega}\) are in I, then there exists a mapping \(f: X\to Y\) such that for every set \(S\subset X\) with \(S\not\in I\), \(f| S\) is not I(S)-continuous, where \(I(S)=\{A\in I:\) \(A\subset S\}\).
Reviewer: A.Błaszczyk

MSC:

54C05 Continuous maps
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] S. Banach, Théorème sur les ensembles de première catégorie, Fund. Math. 16 (1930), 395-398. · JFM 56.0846.04
[2] Adam Emeryk, Ryszard Frankiewicz, and Włdysław Kulpa, On functions having the Baire property, Bull. Acad. Polon. Sci. Sér. Sci. Math. 27 (1979), no. 6, 489 – 491 (English, with Russian summary). Adam Emeryk, Ryszard Frankiewicz, and Władysław Kulpa, Remarks on Kuratowski’s theorem on meager sets, Bull. Acad. Polon. Sci. Sér. Sci. Math. 27 (1979), no. 6, 493 – 498 (English, with Russian summary).
[3] K. Kuratowski, La propriété de Baire dans les espaces métriques, Fund. Math. 16 (1930), 390-394. · JFM 56.0846.03
[4] K. Kuratowski, Topology. Vol. I, New edition, revised and augmented. Translated from the French by J. Jaworowski, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe, Warsaw, 1966. · Zbl 0158.40901
[5] Kazimierz Kuratowski and Andrzej Mostowski, Set theory, Second, completely revised edition, North-Holland Publishing Co., Amsterdam-New York-Oxford; PWN — Polish Scientific Publishers, Warsaw, 1976. With an introduction to descriptive set theory; Translated from the 1966 Polish original; Studies in Logic and the Foundations of Mathematics, Vol. 86. · Zbl 0337.02034
[6] John C. Oxtoby, Measure and category, 2nd ed., Graduate Texts in Mathematics, vol. 2, Springer-Verlag, New York-Berlin, 1980. A survey of the analogies between topological and measure spaces. · Zbl 0435.28011
[7] W. Sierpiński and A. Zygmund, Sur une fonction qui est discontinue sur tout ensemble de puissance du continu, Fund. Math. 4 (1923), 316-318. · JFM 49.0179.02
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.