According to the celebrated theorem of Comfort and Ross (1966), the product of an arbitrary family of pseudocompact topological groups is pseudocompact. We present an overview of several important generalizations of this result, both of “absolute” and “relative” nature. One of them is the preservation of functional boundedness for subsets of topological groups. Also we consider close notions of...

It was shown by Dikranjan and Shakhmatov in 1992 that if a compact abelian group K admits a proper totally dense pseudocompact subgroup, then K cannot have a torsion closed Gδ-subgroup; moreover this condition was shown to be also sufficient under LH. We prove in ZFC that this condition actually ensures the existence of a proper totally dense subgroup H of K that contains an ω-bounded dense sub...

It is known that if P is either the property w-bounded or countably compact, then for every cardinal a 2 w there is a P-group G such that H.G = a and no proper, dense subgroup of G is a P-group. What happens when P is the property pseudocompact? The first-listed author and Robertson have shown that every zero-dimensional Abelian P-group G with H.G > o has a proper, dense, P-group. Turning to th...

A Tychonoff space $X$ is called $\kappa$-pseudocompact if for every continuous mapping $f$ of into $\mathbb{R}^\kappa$ the image $f(X)$ compact. This notion generalizes pseudocompactness and gives a stratification spaces lying between pseudocompact compact spaces. It well known that determined by uniform structure function $C_p(X)$ real-valued functions on endowed with pointwise topology. In re...

A Hausdorff topological group G is minimal if every continuous isomorphism f : G → H between G and a Hausdorff topological group H is open. Significantly strengthening a 1981 result of Stoyanov we prove the following theorem: For every infinite minimal group G there exists a sequence {σn : n ∈ N} of cardinals such that w(G) = sup{σn : n ∈ N} and sup{2 σn : n ∈ N} ≤ |G| ≤ 2, where w(G) is the we...