ar X iv : 0 90 2 . 22 58 v 1 [ m at h . G N ] 1 3 Fe b 20 09 Locally precompact groups : ( Local ) realcompactness and connectedness *

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: (a) Dieudonné completeness; (b) local realcompactness; (c) realcompactness; (d) hereditary realcompactness; (e) connectedness; (f) local connectedness. 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. A subset X of a topological group G is precompact if for every neighborhood U of the identity in G, there is a finite S ⊆ X such that X ⊆ (SU)∩(US). It is easily seen that a subgroup G of a compact group is precompact. The local version of that statement, and its converse, are the content of a theorem of A. Weil: A topological group G embeds as a (dense) subgroup of a locally compact group G if and only if G is locally precompact in the sense that some non-empty open subset of G is precompact (cf. [63]). When such a G is given, the (two-sided) completion G, the Ra˘ ıkov completion of G, is unique in the obvious sense (cf. [48]). As the large bibliographies in the monographs [33], [34], [35], and [42] attest, there is a huge literature devoted to the study and characterization of the locally compact groups that enjoy additional special topological properties. We work here in a parallel vein, but now in the class of locally 1 The first author gratefully acknowledges generous hospitality from the University of Manitoba in January 2008. 2 The second author gratefully acknowledges the generous financial support received from NSERC and the University of Manitoba, which enabled him to do this research.