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Keywords:
space of subgroups; cellularity; Shanin number
space of subgroups; cellularity; Shanin number
Summary:
Given a discrete group $G$, we consider the set $\Cal L(G)$ of all subgroups of $G$ endowed with topology of pointwise convergence arising from the standard embedding of $\Cal L(G)$ into the Cantor cube $\{0,1\}^{G}$. We show that the cellularity $c(\Cal L(G))\leq \aleph_0$ for every abelian group $G$, and, for every infinite cardinal $\tau$, we construct a group $G$ with $c(\Cal L(G))=\tau$.
References:
[1] Engelking R.: General Topology. PWN, Warszawa, 1985. Zbl 0684.54001
[2] Protasov I.V.: Dualisms of topological Abelian groups. Ukrain. Mat. Zh. 31 (1979), 207-211; English translation: Ukrainian Math. J. 31 (1979), 164-166. DOI 10.1007/BF01086015 | MR 0530080 | Zbl 0428.22001
[3] Protasov I.V.: The Suslin number of a space of subgroups of a locally compact group. Ukrain. Mat. Zh. 40 (1988), 654-658; English translation: Ukrainian Math. J. 40 (1988), 559-562. DOI 10.1007/BF01057543 | MR 0971739
[4] Shanin N.A.: On product of topological spaces. Trudy Mat. Inst. Akad. Nauk SSSR 24 (1948), 1-112 (in Russian). MR 0027310
[5] Tsybenko Yu.V.: Dyadicity of a space of subgroups of a topological group. Ukrain. Mat. Zh. 38 (1986), 635-639. MR 0870369
[6] Fuchs L.: Infinite Abelian Groups. Vol. 1, Academic Press, New York and London, 1970. MR 0255673 | Zbl 0338.20063
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