The Chabauty topology on the lattice of closed subgroups (Q1063699)

For a locally compact group G, let \({\mathcal L}(G)\) be the lattice of all closed subgroups of G endowed with the Chabauty topology. There exist two natural operations \(\bigvee\) and \(\bigwedge\) on the lattice \({\mathcal L}(G)\), where \(A\bigvee B\) is the minimal closed subgroup of G generated by \(A\cup B\) and \(A\bigwedge B=A\cap B\), for any A,B\(\in {\mathcal L}(G).\) The aim of the paper is to characterize the classes of \(\bigwedge\)-groups and \(\bigvee\)-groups with continuous operations \(\bigwedge\) and \(\bigvee\), respectively. Primary this characterization is given for compact \(\bigwedge\)-groups (Lemma 6). Then it is shown that a locally compact group G is a \(\bigwedge\)-group iff any compact subgroup of G is a \(\bigwedge\)-group, Cent(g) is open for any topological p-element \(g\in G\) and G is discrete or periodic (Theorem 1). Theorem 2 characterizes those groups which are \(\bigwedge\)-groups and \(\bigvee\)-groups simultaneously.