There are numerous statements in various areas of mathematics (algebra, analysis, geometry, …) that are equivalent to the continuum hypothesis (CH). The earliest instance of this phenomenon is Sierpiński’s theorem from 1919: CH is equivalent to the existence of two sets covering the plane such that every horizontal line has countable intersection with the first set and every vertical line has countable intersection with the second. Sierpiński’s theorem is the blueprint for most other geometric facts that are equivalent to CH. I will survey some of these theorems proved in the last hundred years, and present some new results in this area.
