Bounded sets in spaces and topological groups

We investigate C-compact and relatively pseudocompact subsets of Tychonoff spaces with a special emphasis given to subsets of topological groups. It is shown that a relatively pseudocompact subset of a space X is C-compact in X, but not vice versa. If, however, X is a topological group, then these properties coincide. A product of two C-compact (relatively pseudocompact) subsets A of X and B of Y need not be C-compact (relatively pseudocompact) in X×Y, but if one of the factors X, Y is a topological group, then both C-compactness and relative pseudocompactness are preserved. We prove under the same assumption that, with A and B being bounded subsets of X and Y, the closure of A×B in υ(X×Y) is naturally homeomorphic to clυXA×clυYB, where υ stands for the Hewitt realcompactification. One of our main technical tools is the notion of an R-factorizable group. We show that an R-factorizable subgroup H of an arbitrary group G is z-embedded in G. This fact is applied to prove that the group operations of an R-factorizable group G can always be extended to the realcompactification υG of G, thus giving to υG the topological group structure. We also prove that a C-compact subset A of a topological group G is relatively pseudocompact in the subspace B=A·A−1·A of G.