# Sierpiński set of Euclidean plane

A subset $S$ of $\mathbb{R}^{2}$ is called a Sierpiński set of the plane, if every line parallel to the $x$-axis intersects $S$ only in countably many points and every line parallel to the $y$-axis avoids $S$ in only countably many points:

 $\{x\in\mathbb{R}\,\vdots\;\,(x,\,y)\in S\}\,\mbox{ is countable for all }y\in% \mathbb{R}$
 $\{y\in\mathbb{R}\,\vdots\;\,(x,\,y)\notin S\}\,\mbox{ is countable for all }x% \in\mathbb{R}$

The existence of Sierpiński sets is equivalent (http://planetmath.org/Equivalent3) with the continuum hypothesis, as is proved in [1].

## References

• 1 Gerald Kuba:  “Wie plausibel ist die Kontinuumshypothese?”.  –Elemente der Mathematik 61 (2006).
Title Sierpiński set of Euclidean plane SierpinskiSetOfEuclideanPlane 2013-05-18 23:13:46 2013-05-18 23:13:46 pahio (2872) unlord (1) 11 pahio (1) Definition msc 03E50 Countable Sierpinski set