This thesis concerns twopoint sets, which are subsets of the real plane which
intersect every line in exactly two points. The existence of twopoint sets was first
shown in 1914 by Mazukiewicz, and since this time, the properties of these objects
have been of great intrigue to mathematicians working in both topology and set
theory.
Arguably, the most famous problem about twopoint sets is concerned with their
socalled "descriptive complexity"; it remains open, and it appears to be deep. An
informal interpretation of the problem, which traces back at least to Erdos, is:
The term "twopoint" set can be defined in a way that it is easily
understood by someone with only a limited amount of mathemat
ical training. Even so, how hard is it to construct a twopoint set?
Can one give an effective algorithm which describes precisely how
to do so?
More formally, Erdos wanted to know if there exists a twopoint set which is a Borel
subset of the plane.
An essential tool in showing the existence of a twopoint set is the Axiom of
Choice, an axiom which is taken to be one of the basic truths of mathematics.
