Sierpiński set of Euclidean plane

A subset $S$ of ${ℝ}^2$ 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:

$$\{x\in ℝ\mathrm{⋮}(x,y)\in S\}\text{ is countable for all }y\in ℝ$$

$$\{y\in ℝ\mathrm{⋮}(x,y)\notin S\}\text{ is countable for all }x\in ℝ$$

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