On rectifiable spaces and paratopological groups

TL;DR

This paper investigates properties of paratopological groups and rectifiable spaces, establishing new results on their cardinal invariants, metrizability, and structural characteristics, and addressing open problems in the field.

Contribution

It provides new insights into the structure and properties of rectifiable spaces and paratopological groups, including answers to open problems and conditions for metrizability.

Findings

$ ext{AB}$ is $ ext{omega}$-narrow if $A$ and $B$ are $ ext{omega}$-narrow subsets.

Every bisequential or weakly first-countable rectifiable space is metrizable.

Rectifiable spaces with certain properties contain no closed copy of $S_{ ext{omega}_1}$.

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.