Journal of Lie Theory 20 (2010), No. 2, 347--374
Copyright Heldermann Verlag 2010
Locally Precompact Groups: (Local) Realcompactness and Connectedness
William W. Comfort
Dept. of Mathematics, Wesleyan University, Middletown, CT 06459, U.S.A.
Dept. of Mathematics, University of Manitoba, Winnipeg, Manitoba R3T 2N2, Canada
A theorem of A. Weil asserts that a topological group embeds as a (dense) subgroup of a locally compact group if and only if it contains a non-empty precompact open set; such groups are called "locally precompact. Within the class of locally precompact groups, the authors classify those groups with the following topological properties:
(1) Dieudonné completeness; (2) local realcompactness; (3) realcompactness; (4) hereditary realcompactness; (5) connectedness; (6) local connectedness; (7) zero-dimensionality.
They also prove that an abelian locally precompact group occurs as the quasi-component of a topological group if and only if it is "precompactly generated", that is, it is generated algebraically by a precompact subset.
Keywords: Precompact group, precompactly generated group, locally precompact group, Weil completion, Dieudonné complete group, locally Dieudonné complete group, realcompact group, locally realcompact group, connected group, locally connected group, omega-balanced g
MSC: 22A05, 54H11; 22B05, 22C05
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