Cellularity of a space of subgroups of a discrete group

Arkady G. Leiderman; Igor V. Protasov

Commentationes Mathematicae Universitatis Carolinae (2008)

  • Volume: 49, Issue: 3, page 519-522
  • ISSN: 0010-2628

Abstract

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Given a discrete group G , we consider the set ℒ ( G ) of all subgroups of G endowed with topology of pointwise convergence arising from the standard embedding of ℒ ( G ) into the Cantor cube { 0 , 1 } G . We show that the cellularity c ( ℒ ( G ) ) ≤ ℵ 0 for every abelian group G , and, for every infinite cardinal τ , we construct a group G with c ( ℒ ( G ) ) = τ .

How to cite

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Leiderman, Arkady G., and Protasov, Igor V.. "Cellularity of a space of subgroups of a discrete group." Commentationes Mathematicae Universitatis Carolinae 49.3 (2008): 519-522. <http://eudml.org/doc/250298>.

@article{Leiderman2008,
abstract = {Given a discrete group $G$, we consider the set $\mathcal \{L\}(G)$ of all subgroups of $G$ endowed with topology of pointwise convergence arising from the standard embedding of $\mathcal \{L\}(G)$ into the Cantor cube $\lbrace 0,1\rbrace ^\{G\}$. We show that the cellularity $c(\mathcal \{L\}(G))\le \aleph _0$ for every abelian group $G$, and, for every infinite cardinal $\tau $, we construct a group $G$ with $c(\mathcal \{L\}(G))=\tau $.},
author = {Leiderman, Arkady G., Protasov, Igor V.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {space of subgroups; cellularity; Shanin number; space of subgroups; cellularity; Shanin number},
language = {eng},
number = {3},
pages = {519-522},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Cellularity of a space of subgroups of a discrete group},
url = {http://eudml.org/doc/250298},
volume = {49},
year = {2008},
}

TY - JOUR
AU - Leiderman, Arkady G.
AU - Protasov, Igor V.
TI - Cellularity of a space of subgroups of a discrete group
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2008
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 49
IS - 3
SP - 519
EP - 522
AB - Given a discrete group $G$, we consider the set $\mathcal {L}(G)$ of all subgroups of $G$ endowed with topology of pointwise convergence arising from the standard embedding of $\mathcal {L}(G)$ into the Cantor cube $\lbrace 0,1\rbrace ^{G}$. We show that the cellularity $c(\mathcal {L}(G))\le \aleph _0$ for every abelian group $G$, and, for every infinite cardinal $\tau $, we construct a group $G$ with $c(\mathcal {L}(G))=\tau $.
LA - eng
KW - space of subgroups; cellularity; Shanin number; space of subgroups; cellularity; Shanin number
UR - http://eudml.org/doc/250298
ER -

References

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  1. Engelking R., General Topology, PWN, Warszawa, 1985. Zbl0684.54001
  2. Protasov I.V., 10.1007/BF01086015, Ukrain. Mat. Zh. 31 (1979), 207-211; English translation: Ukrainian Math. J. 31 (1979), 164-166. (1979) Zbl0428.22001MR0530080DOI10.1007/BF01086015
  3. Protasov I.V., 10.1007/BF01057543, Ukrain. Mat. Zh. 40 (1988), 654-658; English translation: Ukrainian Math. J. 40 (1988), 559-562. (1988) MR0971739DOI10.1007/BF01057543
  4. Shanin N.A., On product of topological spaces, Trudy Mat. Inst. Akad. Nauk SSSR 24 (1948), 1-112 (in Russian). (1948) MR0027310
  5. Tsybenko Yu.V., Dyadicity of a space of subgroups of a topological group, Ukrain. Mat. Zh. 38 (1986), 635-639. (1986) MR0870369
  6. Fuchs L., Infinite Abelian Groups, Vol. 1, Academic Press, New York and London, 1970. Zbl0338.20063MR0255673

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