Let G be a locally compact group, and let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">${\mathcal {L}_v}\hspace{-0.6pt}\left( G\right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">L</mi> <mi>v</mi> </msub> <mspace width="-0.6pt" /> <mfenced close=")" open="("> <mi>G</mi> </mfenced> </mrow> </math></EquationSource> </InlineEquation> denote the space of closed subgroups of G, endowed with the Vietoris topology. We define the map <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\delta _{G}:G\rightarrow {\mathcal {L}_v}\hspace{-0.6pt}\left( G\right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>δ</mi> <mi>G</mi> </msub> <mo>:</mo> <mi>G</mi> <mo stretchy="false">→</mo> <msub> <mi mathvariant="script">L</mi> <mi>v</mi> </msub> <mspace width="-0.6pt" /> <mfenced close=")" open="("> <mi>G</mi> </mfenced> </mrow> </math></EquationSource> </InlineEquation>, which assigns to each element <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$g\in G$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>g</mi> <mo>∈</mo> <mi>G</mi> </mrow> </math></EquationSource> </InlineEquation> the closure of the cyclic subgroup generated by g, denoted <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\overline{\textrm{gp}}\left( g\right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover> <mtext>gp</mtext> <mo>¯</mo> </mover> <mfenced close=")" open="("> <mi>g</mi> </mfenced> </mrow> </math></EquationSource> </InlineEquation>. It is natural to ask for which groups G the map <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\delta _{G}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>δ</mi> <mi>G</mi> </msub> </math></EquationSource> </InlineEquation> is continuous. In this paper, we show that <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\delta _{G}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>δ</mi> <mi>G</mi> </msub> </math></EquationSource> </InlineEquation> is continuous at the identity element if and only if G is totally disconnected. Furthermore, we prove that if G is a metrizable, totally disconnected, locally compact group, then <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\delta _{G}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>δ</mi> <mi>G</mi> </msub> </math></EquationSource> </InlineEquation> is continuous on G if and only if G is either discrete or periodic. These results complement and extend the previous work of Hofmann and Willis, as well as Hamrouni and Kammoun, on the continuity of the same map with respect to the Chabauty topology.

Cyclic Subgroups and Vietoris Topology of a Locally Compact Group

期刊论文
发表日期：

2026年7月29日

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Nejeh Alaya,
Zouhour Jlali

摘要

Let G be a locally compact group, and let ${\mathcal {L}_v}\hspace{-0.6pt}\left( G\right) $ $L_{v} (G)$ denote the space of closed subgroups of G, endowed with the Vietoris topology. We define the map $\delta _{G}:G\rightarrow {\mathcal {L}_v}\hspace{-0.6pt}\left( G\right) $ $δ_{G} : G \to L_{v} (G)$ , which assigns to each element $g\in G$ $g \in G$ the closure of the cyclic subgroup generated by g, denoted $\overline{\textrm{gp}}\left( g\right) $ $\bar{gp} (g)$ . It is natural to ask for which groups G the map $\delta _{G}$ $δ_{G}$ is continuous. In this paper, we show that $\delta _{G}$ $δ_{G}$ is continuous at the identity element if and only if G is totally disconnected. Furthermore, we prove that if G is a metrizable, totally disconnected, locally compact group, then $\delta _{G}$ $δ_{G}$ is continuous on G if and only if G is either discrete or periodic. These results complement and extend the previous work of Hofmann and Willis, as well as Hamrouni and Kammoun, on the continuity of the same map with respect to the Chabauty topology.