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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Given an increasing sequence (Xn)nω 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)nω 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)nω. Moreover, some questions about direct limits of qu-lim Xn and pg-lim Xn are posed.

Article information

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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]

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


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.

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