On the space of subgroups of a compact group II

Let G be a compact group. Let S ( G ) , C ( G ) , N ( G ) be the spaces of closed subgroups, cosets of closed subgroups, normal closed subgroups (respectively) of G, with the Vietoris topology. (1) S ( G ) and C ( G ) are never connected; (2) N ( G ) is always totally disconnected; (3) C ( G ) is totally disconnected if and only if G is totally disconnected; and (4) S ( G ) is totally disconnected if and only if G / Z ( G ) is totally disconnected. r: for totally disconnected G (equivalently, profinite G) (5) S ( G ) , C ( G ) and N ( G ) are κ-metrisable; (6) S ( G ) , C ( G ) and N ( G ) are Dugundji compact if G has small weight; and (7) consequences for field extensions are derived.