A note on the WAP-compactification and the LUC-compactification of a topological group

Let G be a topological group. Denote by G(LUC) and G(WAP) the LUC-compactification and the WAP-compactification of G, respectively. G(WAP) can be regarded as a quotient of G(LUC) and the quotient map denoted by pi. In this note we shall show that, when G is a SIN-group, there exists a dense open subset of G(LUC)\ G, consisting of points of unicity for pi, of cardinality at least 2(2kappa (G)), where kappa (G) denotes the compact covering number of G. We give an example to show that this statement does not hold for IN-groups, although G(LUC)\ G does contain at least 2(2kappa (G)) points if G is an IN-group. We also give characterisations of the completion (G) over tilde of G as a subspace of the uniform compactification uG. A consequence of the first result is an analogue of Veech's theorem for the WAP-compactification of a SIN-group.