On rectifiable spaces and paratopological groups

Abstract

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 $G$, which give an affirmative answer for \cite[Open problem 5.1.9]{A2008}; (2) Every bisequential or weakly first-countable rectifiable space is metrizable; (3) The properties of Fr$\acute{e}$chet-Urysohn and strongly Fr$\acute{e}$chet-Urysohn are 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 closed $k$-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 give a partial answer to questions posed by C. Liu in \cite{Liu2009} and C. Liu, S. Lin in \cite{Liu20091}, respectively.