The countability index of the endomorphism semigroup of a vector space

The countability index C(S) of a semigroup S is the least positive integer n, if such an integer exists, with the property that every countable subset of S is contained in a subsemigroup with n generators. If no such integer exists. C(S) is defined to be infinite. Let V be a vector space over a field F and denote by End V the endomorphism semigroup of V. In the two main results, it is determined precisely when C(End V)=2 and when C(End V)=x SpecificallyC(End V)=2 if and only if V is infinite dimensional or dim V=1 and F is finite and C(End V)=x if and only if F is infinite and dim V is an integer N≥1.