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 ω-narrow subsets of a paratopological group G, then AB is ω-narrow in G, which gives an affirmative answer for A.V. Arhangelʹshiı̌ 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 Fréchet–Urysohn and strongly Fréchet–Urysohn coincide in rectifiable spaces; (4) Every rectifiable space G contains a (closed) copy of S ω if and only if G has a (closed) copy of S 2 ; (5) If a rectifiable space G has a σ-point-discrete k-network, then G contains no closed copy of S ω 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.