countable algebraic sets
Proof: Let be a countably infinite subset of . By a cardinality argument (see the attachment), there must exist a line such that the projection of this set to the line is infinite. Since the projection of an algebraic set to a linear subspace is an algebraic set, the projection of 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 polynomial, hence must be finite. Therefore, could not be algebraic since that would lead to a contradiction.
|Title||countable algebraic sets|
|Date of creation||2013-03-22 15:44:41|
|Last modified on||2013-03-22 15:44:41|
|Last modified by||rspuzio (6075)|