Direct limit topologies of quasi-uniform spaces and paratopological groups

Abstract

Given an increasing sequence $(X_n{)}_{n\in \omega }$ of quasi-uniform spaces and paratopological groups, we study the topology of the direct limits $\mathrm{qu}$-$\underrightarrow{lim\hspace{0.17em}}{X}_n$ and $\mathrm{pg}$-$\underrightarrow{lim\hspace{0.17em}}{X}_n$ of the sequence $(X_n{)}_{n\in \omega }$ in the categories of quasi-uniform spaces and paratopological groups, respectively. First, we prove that the quasi-uniformity of the quasi-uniform direct limit $\mathrm{qu}$-$\underrightarrow{lim\hspace{0.17em}}{X}_n$ is generated by some special family of quasi-pseudometrics. Then we discuss some properties of the direct limits $\mathrm{pg}$-$\underrightarrow{lim\hspace{0.17em}}{X}_n$. Finally, we give an explicit description of the topology of the direct limit $\mathrm{pg}$-$\underrightarrow{lim\hspace{0.17em}}{X}_n$ under certain conditions on the sequence of paratopological groups $(X_n{)}_{n\in \omega }$. Moreover, some questions about direct limits of $\mathrm{qu}$-$\underrightarrow{lim\hspace{0.17em}}{X}_n$ and $\mathrm{pg}$-$\underrightarrow{lim\hspace{0.17em}}{X}_n$ are posed.