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The set of discontinuity points of separately continuous functions on the products of compact spaces

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Abstract

We solve the problem of constructing separately continuous functions on the product of compact spaces with a given set of discontinuity points. We obtain the following results. 1. For arbitrary \v{C}ech complete spaces $X$, $Y$, and a separable compact perfect projectively nowhere dense zero set $E\subseteq X\times Y$ there exists a separately continuous function $f:X\times Y\to\mathbb R$ the set of discontinuity points, which coincides with $E$. 2. For arbitrary \v{C}ech complete spaces $X$, $Y$, and nowhere dense zero sets $A\subseteq X$ and $B\subseteq Y$ there exists a separately continuous function $f:X\times Y\to\mathbb R$ such that the projections of the set of discontinuity points of $f$ coincides with $A$ and $B$, respectively. We construct an example of Eberlein compacts $X$, $Y$, and nowhere dense zero sets $A\subseteq X$ and $B\subseteq Y$ such that the set of discontinuity points of every separately continuous function $f:X\times Y\to\mathbb R$ does not coincide with $A\times B$, and a $CH$-example of separable Valdivia compacts $X$, $Y$ and separable nowhere dense zero sets $A\subseteq X$ and $B\subseteq Y$ such that the set of discontinuity points of every separately continuous function $f:X\times Y\to\mathbb R$ does not coincide with $A\times B$.


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Article Information

TitleThe set of discontinuity points of separately continuous functions on the products of compact spaces
SourceMethods Funct. Anal. Topology, Vol. 13 (2007), no. 3, 284-295
MathSciNet MR2356761
CopyrightThe Author(s) 2007 (CC BY-SA)

Authors Information

V. Mykhaylyuk
Department of Mathematical Analysis, Chernivtsi National University, 2 Kotsjubynskoho, Chernivtsi, 58012, Ukraine


Citation Example

V. Mykhaylyuk, The set of discontinuity points of separately continuous functions on the products of compact spaces, Methods Funct. Anal. Topology 13 (2007), no. 3, 284-295.


BibTex

@article {MFAT423,
    AUTHOR = {Mykhaylyuk, V.},
     TITLE = {The set of discontinuity points of separately continuous functions on the products of compact spaces},
   JOURNAL = {Methods Funct. Anal. Topology},
  FJOURNAL = {Methods of Functional Analysis and Topology},
    VOLUME = {13},
      YEAR = {2007},
    NUMBER = {3},
     PAGES = {284-295},
      ISSN = {1029-3531},
       URL = {http://mfat.imath.kiev.ua/article/?id=423},
}


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