  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.
