On the space of subgroups of a compact group. I
[This paper is preparatory to, and should be read prior to, the companion paper ``On the space of subgroups of a compact group. II'', although inexplicably ``I'' follows ``II'' in the printed issue of Topology and its Applications.] Given a compact topological group \(G\) (always Hausdorff here), the authors study in detail these sets of subspaces of \(G\), all in the usual Vietoris topology: \(\mathcal S(G)\), the set of closed subgroups of \(G\); \(\mathcal C(G)\), the set of cosets \(xH\), with \(x\in G\), \(H\in\mathcal S(G)\); and \(\mathcal N(G)\), the set of closed, normal subgroups of \(G\). Among the authors' several results are these: {\parindent=7mm \begin{itemize}\item[(1)] \(\mathcal S(G)\), \(\mathcal C(G)\), and \(\mathcal N(G)\) are compact Hausdorff spaces, and \(\mathcal S\), \(\mathcal C\) and \(\mathcal N\) are covariant functors from the category of compact Hausdorff groups to the category of compact Hausdorff spaces. \item[(2)] \(\mathcal S(G)\) is a retract of \(\mathcal C(G)\). \item[(3)] \(\mathcal S(G)\) is naturally homeomorphic to the orbit space \(\mathcal C(G)/G\). \item[(4)] The topological characters \(\chi(I,\mathcal S(G))\) and \(\chi(1_G,G)\) are equal, where \(I=\{1_G\}\in\mathcal S(G)\). \item[(5)] For \(H\in\mathcal S(G)\), the topology of \(\mathcal S(H)\) is the topology inherited from \(\mathcal S(G)\supseteq\mathcal S(H)\). \item[(6)] For compact \(G_1\), \(G_2\), the map \(h:\mathcal S(G_1)\times\mathcal S(G_1)\rightarrow\mathcal S(G_1\times G_2)\) defined by \(h(H_1,H_2)=H_1\times H_2\) is surjective (hence, a homeomorphism) iff \(G_1\) and \(G_2\) are profinite (=~totally disconnected) and \(\Sigma(G_1)\cap\Sigma(G_2)=\emptyset\); here, \(\Sigma(G)\) is the set of primes \(p\) for which \(G\) contains a topological copy of either \(\mathbb Z_p\) or \(C_{p^n}\) for some \(n\geq1\). \end{itemize}}
scientific article; zbMATH DE number 5539509
scientific article