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

TL;DR

This paper investigates the diversity of topologies on infinite Abelian groups, showing a vast number of nonequivalent bounded Hausdorff group topologies and characterizing compact topologies on a specific group under the continuum hypothesis.

Contribution

It establishes the exact number of nonequivalent bounded Hausdorff topologies on infinite Abelian groups and determines the compact topologies on $( ext{Z}_p)^{ ext{N}}$ under the continuum hypothesis.

Findings

Existence of $2^{2^{f m}}$ nonequivalent bounded Hausdorff topologies on any infinite Abelian group.

Under continuum hypothesis, the number of compact topologies on $( ext{Z}_p)^{ ext{N}}$ is determined.

Characterization of topologies on $( ext{Z}_p)^{ ext{N}}$ under set-theoretic assumptions.

Abstract

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