A common shorthand is to denote "D"("Sf") by "D"("f"), where "Sf" is the ideal generated by "f". For any ideal "a", the sets "D"("a") and "V"("a") are complementary, and hence the same proof as before shows that the sets "D"("a") form a topology on formula_7. The advantage of this approach is that the sets "D"("f"), where "f" ranges over all homogeneous elements of the ring "S", form a base for this topology, which is an indispensable tool for the analysis of formula_7, just as the analogous fact for the spectrum of a ring is likewise indispensable.