Vol. 63, No. 1, 1976

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Spaces of discrete subsets of a locally compact group

Norman Oler

Vol. 63 (1976), No. 1, 213–219

This paper represents a continuing effort to develop elements of a general theory of packing and covering by translates of a fixed subset of a group. For P a subset of a group X a subset L is a left P-packing if for any distinct elements x1 and x2 in L, x1P x2P is empty. A subset M of X is a left P-covering if MP = X. The Chabauty topology on the set of discrete subgroups of a locally compact group has been used only to a rather limited extent in the Geometry of Numbers but by its very definition is a natural one to work with in studying packing and covering problems. The main results of this paper are that the Chabauty topology extends to the family of all closed discrete subsets containing the identity and that if X is σ-compact then S(X,P) the space of all left P-packings for a fixed neighborhood P of the identity is locally compact.

Mathematical Subject Classification 2000
Primary: 22D99
Secondary: 22E40, 10E30
Received: 31 January 1975
Revised: 17 September 1975
Published: 1 March 1976
Norman Oler