Banach Journal of Mathematical Analysis

Direct limit topologies of quasi-uniform spaces and paratopological groups

Fucai Lin, Li-Hong Xie, and Jing Zhang

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Abstract

Given an increasing sequence (Xn)n∈ω of quasi-uniform spaces and paratopological groups, we study the topology of the direct limits qu-lim →Xn and pg-lim →Xn of the sequence (Xn)n∈ω in the categories of quasi-uniform spaces and paratopological groups, respectively. First, we prove that the quasi-uniformity of the quasi-uniform direct limit qu-lim →Xn is generated by some special family of quasi-pseudometrics. Then we discuss some properties of the direct limits pg-lim →Xn. Finally, we give an explicit description of the topology of the direct limit pg-lim →Xn under certain conditions on the sequence of paratopological groups (Xn)n∈ω. Moreover, some questions about direct limits of qu-lim →Xn and pg-lim →Xn are posed.

Article information

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

Dates
Received: 26 June 2014
Accepted: 7 April 2015
First available in Project Euclid: 15 October 2015

Permanent link to this document
https://projecteuclid.org/euclid.bjma/1444913862

Digital Object Identifier
doi:10.1215/17358787-3163248

Mathematical Reviews number (MathSciNet)
MR3453523

Zentralblatt MATH identifier
1335.22003

Subjects
Primary: 46A13: Spaces defined by inductive or projective limits (LB, LF, etc.) [See also 46M40]
Secondary: 22A05: Structure of general topological groups 54B30: Categorical methods [See also 18B30] 54E15: Uniform structures and generalizations 54H11: Topological groups [See also 22A05]

Keywords
direct limit quasi-uniform spaces paratopological groups quasi-pseudometric semitopological groups

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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