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Register a new userTopologies on groups determined by sequences: Answers to several questions of I.Protasov and E.Zelenyuk
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Taras Banakh
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We answer several questions of I.Protasov and E.Zelenyuk concerning topologies on groups determined by T-sequences. A special attention is paid to studying the operation of supremum of two group topologies.
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We introduce and study oscillator topologies on paratopological groups and define certain related number invariants. As an application we prove that a Hausdorff paratopological group $G$ admits a weaker Hausdorff group topology provided $G$ is 3-oscillating. A paratopological group $G$ is 3-oscillating (resp. 2-oscillating) provided for any neighborhood $U$ of the unity $e$ of $G$ there is a neighborhood $Vsubset G$ of $e$ such that $V^{-1}VV^{-1}subset UU^{-1}U$ (resp. $V^{-1}Vsubset UU^{-1}$). The class of 2-oscillating paratopological groups includes all collapsing, all nilpotent paratopological groups, all paratopological groups satisfying a positive law, all paratopological SIN-group and all saturated paratopological groups (the latter means that for any nonempty open set $Usubset G$ the set $U^{-1}$ has nonempty interior). We prove that each totally bounded paratopological group $G$ is countably cellular; moreover, every cardinal of uncountable cofinality is a precaliber of $G$. Also we give an example of a saturated paratopological group which is not isomorphic to its mirror paratopological group as well as an example of a 2-oscillating paratopological group whose mirror paratopological group is not 2-oscillating.
(1) Every infinite, Abelian compact (Hausdorff) group K admits 2^|K|-many dense, non-Haar-measurable subgroups of cardinality |K|. When K is nonmetrizable, these may be chosen to be pseudocompact. (2) Every infinite Abelian group G admits a family A of 2^2^|G|-many pairwise nonhomeomorphic totally bounded group topologies such that no nontrivial sequence in G converges in any of the topologies T in A. (For some G one may arrange w(G,T) < 2^|G| for some T in A.) (3) Every infinite Abelian group $G$ admits a family B of 2^2^|G|-many pairwise nonhomeomorphic totally bounded group topologies, with w(G,T) = 2^|G| for all T in B, such that some fixed faithfully indexed sequence in G converges to 0_G in each T in B.
We present a number of examples to illustrate the use of small quotient dessins as substitutes for their often much larger and more complicated Galois (minimal regular) covers. In doing so we employ several useful group-theoretic techniques, such as the Frobenius character formula for counting triples in a finite group, pointing out some common traps and misconceptions associated with them. Although our examples are all chosen from Hurwitz curves and groups, they are relevant to dessins of any type.
A theorem of A. Weil asserts that a topological group embeds as a (dense) subgroup of a locally compact group if and only if it contains a non-empty precompact open set; such groups are called locally precompact. Within the class of locally precompact groups, the authors classify those groups with the following topological properties: Dieudonne completeness; local realcompactness; realcompactness; hereditary realcompactness; connectedness; local connectedness; zero-dimensionality. They also prove that an abelian locally precompact group occurs as the quasi-component of a topological group if and only if it is precompactly generated, that is, it is generated algebraically by a precompact subset.
An embeddability criterion for zero-dimensional metrizable topological spaces in zero-dimensional metrizable topological groups is given. A space which can be embedded as a closed subspace in a zero-dimensional metrizable group but is not strongly zero-dimensional is constructed; thereby, an example of a metrizable group with noncoinciding dimensions ind and dim is obtained. It is proved that one of Kuleszas zero-dimensional metrizable spaces cannot be embedded in a metrizable zero-dimensional group.
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