Compact families of noncompact subgroups and clopen partial orders

Abstract

For a locally compact Hausdorff group \(G\), let \(\mathcal N(G)\) be the space of closed noncompact subgroups with the full Vietoris topology. We prove that a compact Hausdorff space \(X\) embeds in some \(\mathcal N(G)\) if and only if \(X\) admits a partial order with clopen principal ideals. In the affirmative case, \(G\) may be chosen to be the discrete abelian group \(\mathbb Z\oplus\bigoplus_X\mathbb Z/2\mathbb Z\). The quotient of \([0,\omega_1]^2\) obtained by collapsing its closed lower triangle is a scattered compact space that admits no such order. It therefore gives a negative answer to Kourovka Problem 9.47.