On Topologies on the Group $(\Z_p)^{\N}$
I.K. Babenko, S.A. Bogatyi

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 , there exist precisely nonequivalent bounded Hausdorff group topologies. Under the continuum hypothesis, the number of nonequivalent compact and locally compact Hausdorff group topologies on the group is determined.
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Taxonomy
TopicsAdvanced Topology and Set Theory · Advanced Banach Space Theory · Mathematical Dynamics and Fractals
