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.
Publié le : 2002-05-14
Classification:
Sierpiński-Zygmund functions and sets,
Continuum Hypothesis,
26A15,
03E50,
03E75
@article{1150118746,
author = {P\l otka, Krzysztof},
title = {The ideal of Sierpinski-Zygmund sets on the plane.},
journal = {Real Anal. Exchange},
volume = {28},
number = {1},
year = {2002},
pages = { 191-198},
language = {en},
url = {http://dml.mathdoc.fr/item/1150118746}
}
Płotka, Krzysztof. The ideal of Sierpinski-Zygmund sets on the plane.. Real Anal. Exchange, Tome 28 (2002) no. 1, pp. 191-198. http://gdmltest.u-ga.fr/item/1150118746/