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Fundamentalnaya i Prikladnaya Matematika, 2015, Volume 20, Issue 2, Pages 105–112 (Mi fpm1643)  

This article is cited in 4 scientific papers (total in 4 papers)

On embeddings of topological groups

S. Iliadis

Lomonosov Moscow State University
Full-text PDF (134 kB) Citations (4)
References:
Abstract: The problem of the existence of universal elements in the class of all topological groups of weight $\leq\tau\neq\omega$ remains open. In this paper, it is proved that for many classes of topological groups there are so-called continuously containing spaces. Let $\mathbb S$ be a saturated class of completely regular spaces of weight $\leq\tau$ and $\mathbb G$ be the subclass of elements of $\mathbb S$ that are topological groups. Then there exists an element $\mathrm T\in\mathbb S$ having the following property: for every $G\in\mathbb T$, there exists a homeomorphism $h^G_\mathrm T$ of $G$ into $\mathrm T$ such that if the points $x,y$ of $\mathrm T$ belong to the set $h^H_\mathrm T(H)$ for some $H\in\mathbb G$, then for every open neighbourhood $U$ of $xy$ in $\mathrm T$ there are open neighbourhoods $V$ and $W$ of $x$ and $y$ in $\mathrm T$, respectively, such that for every $G\in\mathbb G$ we have
$$ \left(V\cap h^G_\mathrm T(G)\right)\left(W\cap h^G_\mathrm T(G)\right)^{-1}\subset U\cap h^G_\mathrm T(G). $$
In this case, we say that $\mathrm T$ is a continuously containing space for the class $\mathbb G$. We recall that as the class $\mathbb S$ we can consider, for example, the following classes of completely regular spaces: $n$-dimensional spaces, countable-dimensional spaces, strongly countable-dimensional spaces, locally finite-dimensional spaces. Therefore, in all these classes there are elements that are continuously containing spaces for the corresponding subclasses consisting of topological groups. In this paper, some open problems are considered.
English version:
Journal of Mathematical Sciences (New York), 2017, Volume 223, Issue 6, Pages 720–724
DOI: https://doi.org/10.1007/s10958-017-3381-9
Bibliographic databases:
Document Type: Article
UDC: 515.12+512.546
Language: Russian
Citation: S. Iliadis, “On embeddings of topological groups”, Fundam. Prikl. Mat., 20:2 (2015), 105–112; J. Math. Sci., 223:6 (2017), 720–724
Citation in format AMSBIB
\Bibitem{Ili15}
\by S.~Iliadis
\paper On embeddings of topological groups
\jour Fundam. Prikl. Mat.
\yr 2015
\vol 20
\issue 2
\pages 105--112
\mathnet{http://mi.mathnet.ru/fpm1643}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=3472271}
\elib{https://elibrary.ru/item.asp?id=25686565}
\transl
\jour J. Math. Sci.
\yr 2017
\vol 223
\issue 6
\pages 720--724
\crossref{https://doi.org/10.1007/s10958-017-3381-9}
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  • https://www.mathnet.ru/eng/fpm1643
  • https://www.mathnet.ru/eng/fpm/v20/i2/p105
  • This publication is cited in the following 4 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Фундаментальная и прикладная математика
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