lemma on projection of countable sets
Suppose is an infinite field and is an infinite subset of . Then there exists a line such that the projection of on is infinite.
Proof: This proof will proceed by an induction on . The case is trivial since a one-dimensional linear space is a line.
Consider two cases:
Case I: There exists a proper subspace of which contains an infinite number of points of .
In this case, we can restrict attention to this subspace. By the induction hypothesis, there exists a line in the subspace such that the projection of points in the subspace to this line is already infinite.
Case II: Every proper subspace of contains at most a finite number of points of .
In this case, any line will do. By definition, one constructs a projection by dropping hyperplanes perpendicular to the line passing through the points of the set. Since each of these hyperplanes will contain a finite number of elements of , an infinite number of hyperplanes will be needed to contain all the points of , hence the projection will be infinite.
|Title||lemma on projection of countable sets|
|Date of creation||2013-03-22 15:44:53|
|Last modified on||2013-03-22 15:44:53|
|Last modified by||rspuzio (6075)|