Sylvester’s problem and Motzkin’s theorem for countable and compact sets

Abstract:

The following three variations of Sylvester’s Problem are established. Let $A$ and $B$ be compact, countable and disjoint sets of points. (1) If $A$ spans ${E^2}$ (the Euclidean plane) then there must exist a line through two points of $A$ that intersects $A$ in only finitely many points. (2) If $A$ spans ${E^3}$ (Euclidean three-space) then there must exist a line through exactly two points of $A$. (3) If $A \cup B$ spans ${E^2}$ then there must exist a line through at least two points of one of the sets that does not intersect the other set.