C-compact and r-pseudocompact subsets of paratopological groups

摘要

Our study of C-compactness, r-pseudocompactness, and close notions is motivated by the fact that an arbitrary product Pi(i is an element of I) B-i of C-compact subsets B-i of respective topological groups G(i) is C-compact in the product group Pi(i is an element of I) G(i), and the same conclusion remains valid for products of r-pseudocompact subsets of topological groups. In fact, it is known that the two notions of boundedness coincide for subsets of topological groups (but they are quite different for subsets of Tychonoff spaces). Our aim here is to extend the aforementioned results to paratopological groups. We find several wide classes of paratopological groups in which the C-compact and r-pseudocompact subsets coincide (these include totally omega-narrow paratopological groups, commutative paratopological groups with countable Hausdorff number, precompact or Lindelof paratopological groups). Similarly, we present several classes of paratopological groups in which C-compact and/or r-pseudocompact subsets remain to be productive, as in topological groups. (C) 2016 Elsevier B.V. All rights reserved.