Let R be a real closed field. The semialgebraic subsets of Rn form the smallest collection of subsets of Rn containing all sets of the form {x 2 Rn| f(x) > 0}, where f 2 R[X1, · · · ,Xn], and closed under complementation and finite union and intersection. The Euclidean topology on Rn is defined by taking the open balls Bn(x, r) := {y 2 Rn| ky −xk M.