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:
$$\{x\in \mathbb{R}\mathrm{\vdots}(x,y)\in S\}\text{is countable for all}y\in \mathbb{R}$$ |
$$\{y\in \mathbb{R}\mathrm{\vdots}(x,y)\notin S\}\text{is countable for all}x\in \mathbb{R}$$ |
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^{} |
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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 | Countable^{} |
Defines | Sierpinski set |