Productive Properties in Topological Groups

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 C-compactness and r-pseudocompactness for subsets of Tychonoff spaces and establish their productivity in the class of topological groups. Finally, we give a very brief overview of productivity properties in paratopological and semitopological groups.