
Journal of Lie Theory 20 (2010), No. 2, 347374 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. wcomfort@wesleyan.edu Gábor Lukács Dept. of Mathematics, University of Manitoba, Winnipeg, Manitoba R3T 2N2, Canada lukacs@cc.umanitoba.ca 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 nonempty 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) zerodimensionality. They also prove that an abelian locally precompact group occurs as the quasicomponent 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, omegabalanced g MSC: 22A05, 54H11; 22B05, 22C05 [ Fulltextpdf (275 KB)] for subscribers only. 