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2002-2003 The ideal of Sierpinski-Zygmund sets on the plane.
Krzysztof Płotka
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Real Anal. Exchange 28(1): 191-198 (2002-2003).


We say that a set $X \sq \real^2$ is {\it Sierpi{\'n}ski-Zygmund\/} (or {\it SZ-set\/} for short) if it does not contain a partial continuous function of cardinality continuum $\cont$. We observe that the family of all such sets is $\cf(\cont)$-additive ideal. Some examples of such sets are given. We also consider {\it SZ-shiftable sets\/}; that is, sets $X \sq \real^2$ for which there exists a function $f\colon \real \to \real$ such that $f+X$ is a SZ-set. Some results are proved about SZ-shiftable sets. In particular, we show that the union of two SZ-shiftable sets does not have to be SZ-shiftable.


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Krzysztof Płotka. "The ideal of Sierpinski-Zygmund sets on the plane.." Real Anal. Exchange 28 (1) 191 - 198, 2002-2003.


Published: 2002-2003
First available in Project Euclid: 12 June 2006

zbMATH: 1052.26003
MathSciNet: MR1973979

Primary: 26A15
Secondary: 03E50 , 03E75

Keywords: Continuum hypothesis , Sierpiński-Zygmund functions and sets

Rights: Copyright © 2002 Michigan State University Press

Vol.28 • No. 1 • 2002-2003
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