countable algebraic sets

An algebraic setMathworldPlanetmath over an uncountably infinite base fieldMathworldPlanetmath 𝔽 (like the real or complex numbersMathworldPlanetmathPlanetmath) cannot be countably infiniteMathworldPlanetmath.

Proof: Let S be a countably infinite subset of 𝔽n. By a cardinality argumentPlanetmathPlanetmath (see the attachment), there must exist a line such that the projection of this set to the line is infiniteMathworldPlanetmathPlanetmath. Since the projection of an algebraic set to a linear subspace is an algebraic set, the projection of S to this line would be an algebraic subset of the line. However, an algebraic subset of a line is the locus of zeros of some polynomialPlanetmathPlanetmath, hence must be finite. Therefore, S could not be algebraic since that would lead to a contradictionMathworldPlanetmathPlanetmath.

Title countable algebraic sets
Canonical name CountableAlgebraicSets
Date of creation 2013-03-22 15:44:41
Last modified on 2013-03-22 15:44:41
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 8
Author rspuzio (6075)
Entry type Theorem
Classification msc 14A10