Sierpiński set of Euclidean plane

A subset S of 2 is called a Sierpiński set of the plane, if every line parallelMathworldPlanetmathPlanetmathPlanetmath 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(x,y)S} is countable for all y
{y(x,y)S} is countable for all x

The existence of Sierpiński sets is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath ( with the continuum hypothesisMathworldPlanetmath, as is proved in [1].


  • 1 Gerald Kuba:  “Wie plausibel ist die Kontinuumshypothese?”.  –Elemente der Mathematik 61 (2006).
Title Sierpiński set of Euclidean planeMathworldPlanetmath
Canonical name SierpinskiSetOfEuclideanPlane
Date of creation 2013-05-18 23:13:46
Last modified on 2013-05-18 23:13:46
Owner pahio (2872)
Last modified by unlord (1)
Numerical id 11
Author pahio (1)
Entry type Definition
Classification msc 03E50
Related topic CountableMathworldPlanetmath
Defines Sierpinski set