On rectifiable spaces and paratopological groups

We mainly discuss the cardinal invariants and generalized metric properties on paratopological groups or rectifiable spaces, and show that: (1) If A and B are omega-narrow subsets of a paratopological group G, then AB is omega-narrow in C. which gives an affirmative answer for A.V. Arhangel'shii and M. Tkachenko (2008) [7. Open problem 5.1.9]; (2) Every bisequential or weakly first-countable rectifiable space is metrizable; (3) The properties of Frechet-Urysohn and strongly Frechet-Urysohn coincide in rectifiable spaces; (4) Every rectifiable space G contains a (closed) copy of S-omega if and only if G has a (closed) copy of S-2: (5) If a rectifiable space G has a sigma-point-discrete k-network, then G contains no closed copy of S-omega 1; (6) If a rectifiable space G is pointwise canonically weakly pseudocompact, then G is a Moscow space. Also, we consider the remainders of paratopological groups or rectifiable spaces, and answer two questions posed by C. Liu (2009) in [20] and C. Liu, S. Lin (2010) in [21], respectively. (C) 2010 Elsevier B.V. All rights reserved.