Topologically independent sets in precompact groups

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F44555601%3A13440%2F18%3A43893065" target="_blank" >RIV/44555601:13440/18:43893065 - isvavai.cz</a>
Result on the web
<a href="http://dx.doi.org/10.1016/j.topol.2017.12.020" target="_blank" >http://dx.doi.org/10.1016/j.topol.2017.12.020</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1016/j.topol.2017.12.020" target="_blank" >10.1016/j.topol.2017.12.020</a>

Result language
angličtina
Original language name
Topologically independent sets in precompact groups
Original language description
It is a simple fact that a subgroup generated by a subset $A$ of an abelian group is the direct sum of the cyclic groups $hull{a}$, $ain A$ if and only if the set $A$ is independent. In cite{DSS} the concept of an {em independent} set in an abelian group was generalized to a {em topologically independent set} in a topological abelian group (these two notions coincide in discrete abelian groups). It was proved that a topological subgroup generated by a subset $A$ of an abelian topological group is the Tychonoff direct sum of the cyclic topological groups $hull{a}$, $ain A$ if and only if the set $A$ is topologically independent and absolutely Cauchy summable. Further, it was shown, that the assumption of absolute Cauchy summability of $A$ can not be removed in general in this result. In our paper we show that it can be removed in precompact groups. In other words, we prove that if $A$ is a subset of a {em precompact} abelian group, then the topological subgroup generated by $A$ is the Tychonoff direct sum of the topological cyclic subgroups $hull{a}$, $ain A$ if and only if $A$ is topologically independent. We show that precompactness can not be replaced by local compactness in this result.
Czech name
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Czech description
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Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
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OECD FORD branch
10101 - Pure mathematics

Project
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Continuities
I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year
2018
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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