Continuity Characterizing Totally Disconnected Locally Compact Groups

For a locally compact group G and its compact space SUB(G) of closed subgroups let mu(G): G -> SUB(G) denote the function which attaches to an element g of G the closed subgroup (< g >) over bar generated by it. It is shown that G is totally disconnected if and only if mu(G) is continuous. Several other functions which associate with an element of G in a natural way a closed subgroup of G are discussed with respect to their continuity in totally disconnected locally compact groups.