Direct limit topologies of quasi-uniform spaces and paratopological groups

Given an increasing sequence $(X_{n})_{n\in\omega}$ of quasi-uniform spaces and paratopological groups, we study the topology of the direct limits $\mathrm{qu}$ - ${\mathop{\lim}\limits_{\longrightarrow}}X_{n}$ and $\mathrm{pg}$ - $\mathop{\lim}\limits_{\longrightarrow}X_{n}$ 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 $\mathrm{qu}$ - $\mathop{\lim}\limits_{\longrightarrow}X_{n}$ is generated by some special family of quasi-pseudometrics. Then we discuss some properties of the direct limits $\mathrm{pg}$ - $\mathop{\lim}\limits_{\longrightarrow}X_{n}$ . Finally, we give an explicit description of the topology of the direct limit $\mathrm{pg}$ - $\mathop{\lim}\limits_{\longrightarrow}X_{n}$ under certain conditions on the sequence of paratopological groups $(X_{n})_{n\in\omega}$ . Moreover, some questions about direct limits of $\mathrm{qu}$ - $\mathop{\lim}\limits_{\longrightarrow}X_{n}$ and $\mathrm{pg}$ - $\mathop{\lim}\limits_{\longrightarrow}X_{n}$ are posed.