abstract
The lattice PC(G) of precompact group topologies on an Abelian group G is isomorphic with the lattice SG(G*) of subgroups of the algebraic character group (Remus, 1983). Remus used this result to determine the number of precompact [Hausdorff] topologies on Abelian groups. In this paper the same tool is applied to the problems of existence and number of maximal precompact [Hausdorff] topologies on an Abelian group G, i.e. antiatoms in the lattice PC(G). It is shown that PC(G) has antiatoms iff G is not torsion-free. Further the number of maximal precompact [Hausdorff] topologies is expressed in terms of the cardinalities of the p-components of the group G.
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References
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Schinkel, F. Existence and Number of Maximal Precompact Topologies on Abelian Groups. Results. Math. 17, 282–286 (1990). https://doi.org/10.1007/BF03322464
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DOI: https://doi.org/10.1007/BF03322464