lemma on projection of countable sets

Suppose 𝔽 is an infinite field and S is an infinite subset of 𝔽n. Then there exists a line L such that the projectionMathworldPlanetmathPlanetmath of S on L is infinite.

Proof: This proof will proceed by an induction on n. The case n=1 is trivial since a one-dimensional linear spacePlanetmathPlanetmath is a line.

Consider two cases:

Case I: There exists a proper subspacePlanetmathPlanetmathPlanetmath of 𝔽n which contains an infinite number of points of S.

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 𝔽n contains at most a finite number of points of S.

In this case, any line will do. By definition, one constructs a projection by dropping hyperplanesMathworldPlanetmath perpendicularMathworldPlanetmathPlanetmathPlanetmath to the line passing through the points of the set. Since each of these hyperplanes will contain a finite number of elements of S, an infinite number of hyperplanes will be needed to contain all the points of S, hence the projection will be infinite.

Title lemma on projection of countable sets
Canonical name LemmaOnProjectionOfCountableSets
Date of creation 2013-03-22 15:44:53
Last modified on 2013-03-22 15:44:53
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 10
Author rspuzio (6075)
Entry type Theorem
Classification msc 14A10