Topologically independent sets in precompact groups

Kód výsledku v 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>

Výsledek na webu

<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>

Jazyk výsledku

angličtina

Název v původním jazyce

Topologically independent sets in precompact groups

Popis výsledku v původním jazyce

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.

Název v anglickém jazyce

Topologically independent sets in precompact groups

Popis výsledku anglicky

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.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

—

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2018

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Topology and its Applications

ISSN

0166-8641

e-ISSN

—

Svazek periodika

2018

Číslo periodika v rámci svazku

235

Stát vydavatele periodika

NL - Nizozemsko

Počet stran výsledku

6

Strana od-do

269-274

Kód UT WoS článku

000426021900020

EID výsledku v databázi Scopus

2-s2.0-85038235161