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It is proved that, on any Abelian group of infinite cardinality ${\bf m}$,there exist precisely $2^{2^{\bf m}}$ nonequivalent bounded Hausdorff grouptopologies. Under the continuum hypothesis, the number of nonequivalent compactand locally compact Hausdorff group topologies on the group $(\Z_p)^{\N}$ isdetermined.

On Topologies on the Group $(\Z_p)^{\N}$