The ideal of Sierpinski-Zygmund sets on the plane.

Abstract

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.