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Zur Struktur des Verbandes der Gruppentopologien. (German) Zbl 0547.22004
Fachbereich Mathematik der Universität Hannover. 92 S. (1983).
The theme of the present thesis is the study of the lattice Tg(G) of group-topologies on a given group G. It is shown that for an infinite abelian group G, Tg(G) is neither complimentary nor distributive. Let PK(G) be the sublattice of the pre-compact topologies (need not be Hausdorff) on G. When G is abelian PK(G) is isomorphic with the lattice of subgroups of the dual group of G. The author also proves that for every infinite abelian group G, there exists exactly \(2^{2^{| G|}}\) Hausdorff precompact topologies (on G). The above results do not hold for general groups. However, the author has obtained the following results: (1) PK(G) is anti-isomorphic with the lattice of the closed normal subgroups of the Bohr-compactification of \((G,\tau_ d)\), here \(\tau_ d\) means discrete topology on G. (2) On every free group F, there exist exactly \(2^{2^{| F|}}\) Hausdorff group- topologies.
Reviewer: T.Wu

MSC:
22D05 General properties and structure of locally compact groups
54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets)