## Banach Journal of Mathematical Analysis

### 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 $\mathrm{qu}$-${\mathop{\lim}\limits_{\longrightarrow}}X_{n}$ 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.

#### Article information

Source
Banach J. Math. Anal., Volume 10, Number 1 (2016), 48-70.

Dates
Accepted: 7 April 2015
First available in Project Euclid: 15 October 2015

https://projecteuclid.org/euclid.bjma/1444913862

Digital Object Identifier
doi:10.1215/17358787-3163248

Mathematical Reviews number (MathSciNet)
MR3453523

Zentralblatt MATH identifier
1335.22003

#### Citation

Lin, Fucai; Xie, Li-Hong; Zhang, Jing. Direct limit topologies of quasi-uniform spaces and paratopological groups. Banach J. Math. Anal. 10 (2016), no. 1, 48--70. doi:10.1215/17358787-3163248. https://projecteuclid.org/euclid.bjma/1444913862

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