The equations x(x-1)=0, y^2+Bxy-y=0 are easy to solve, but the Newton Map

N: C^2 \rightarrow C^2 for finding the four roots has very complicated dynamics: N is four-to-one and N has points of indeterminacy. Furthermore, high iterates of N have many points of indeterminacy. By restricting to parameters |1-B|>1, all of these points of indeterminacy are in the set X_l = Re(x) < 1/2, which is invariant under N. If one wants to consider the homotopy type of a basin of attraction, W(r_i), for one of the roots r_i \in X_l under N, one encounters a kind of ``topological indeterminacy.'' When studying the homotopy type of a loop \gamma in W(r_i), should one consider the homotopies that hit the points of indeterminacy of N^k or should one avoid them? Both seem reasonable. To avoid such questions, one can perform blow-ups at the points of indeterminacy of all iterates of N, obtaining a new space X_l^\infty from X_l on which all iterates of N are defined.