Subsets of ℝ n
Krzysztof Ciesielski · Cambridge University Press eBooks · 1997
Strange subsets of ℝ n and the diagonalization argument In this section we will illustrate some typical constructions by transfinite induction. We will do so by constructing recursively some subsets of ℝ n with strange geometric properties. The choice of geometric descriptions of these sets is not completely arbitrary here – we still have not developed the basic facts concerning “nice” subsets of ℝ n that are necessary for most of our applications. This will be done in the remaining sections of this chapter. To state the next theorem, we will need the following notation. For a subset A of the plane ℝ 2 the horizontal section of A generated by y ∈ ℝ (or, more precisely, its projection onto the first coordinate) will be denoted by A y and defined as A y = { x ∈ ℝ: 〈 x,y 〉 ∈ A }. The vertical section of A generated by x ∈ ℝ is defined by A x = { y ∈ ℝ: 〈 x,y 〉 ∈ A }. We will start with the following example. Theorem 6.1.1 There exists a subset A of the plane with every horizontal section A y being dense in ℝ and with every vertical section A x having precisely one element . Proof We will define the desired set by induction. To do so, we will first reduce our problem to the form that is most appropriate for a recursive construction. The requirement that every horizontal section of A is dense in ℝ tells us that A is “reasonably big.”