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 Universität, Schlossgartenstr. 7, 64289 Darmstadt, Germany
hofmann@mathematik.tu-darmstadt.de
George A. Willis
School of Mathematical and Physical Sciences, University of Newcastle, Callaghan, NSW 2308, Australia
george.willis@newcastle.edu.au
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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