Cellular generators.

The aim of this paper is twofold. On the one hand, we show that the kernel $\overline{C(A)}$ of the Bousfield periodization functor $P_A$ is cellularly generated by a space $B$, i.e., we construct a space $B$ such that the smallest closed class $C(B)$ containing $B$ is exactly $\overline{C(A)}$. On the other hand, we show that the partial order $(Spaces,\gg)$ is a complete lattice, where $B\gg A$ if $B\in C(A)$. Finally, as a corollary we obtain Bousfield's theorem, which states that $(Spaces,>)$ is a complete lattice, where $B>A$ if $B\in\overline{C(A)}$. [ABSTRACT FROM AUTHOR]