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Journal of Lie Theory 25 (2015), No. 1, 001--007
Copyright Heldermann Verlag 2015

Continuity Characterizing Totally Disconnected Locally Compact Groups

Karl H. Hofmann
Fachbereich Mathematik, Technische Universitšt, Schlossgartenstr. 7, 64289 Darmstadt, Germany

George A. Willis
School of Mathematical and Physical Sciences, University of Newcastle, Callaghan, NSW 2308, Australia

For a locally compact group G and its compact space SUB(G) of closed subgroups let μG: G -> SUB(G) denote the function which attaches to an element g of G the closed subgroup generated by it. It is shown that G is totally disconnected if and only if μ 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.

Keywords: Locally compact group, Chabauty space, hyperspace of closed subgroups, continuity, monothetic subgroup.

MSC: 22D05, 22C05, 54B20

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