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) Dieudonne completeness; (b) local realcompactness; (c) realcompactness; (d) hereditary realcompactness, (e) connectedness, (f) local connectedness, (g) 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.