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Local Yoneda completions of quasi-metric spaces

Published online by Cambridge University Press:  24 March 2023

Jing Lu Open the ORCID record for Jing Lu [Opens in a new window]  and
Bin Zhao
Jing Lu
Affiliation:
School of Mathematics and Statistics, Shaanxi Normal University, Xi’an 710119, P.R. China
Bin Zhao*
Affiliation:
School of Mathematics and Statistics, Shaanxi Normal University, Xi’an 710119, P.R. China
*
*Corresponding author. Email: zhaobin@snnu.edu.cn
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Abstract

In this paper, we study quasi-metric spaces using domain theory. Given a quasi-metric space (X,d), we use $({\bf B}(X,d),\leq^{d^{+}}\!)$ to denote the poset of formal balls of the associated quasi-metric space (X,d). We introduce the notion of local Yoneda-complete quasi-metric spaces in terms of domain-theoretic properties of $({\bf B}(X,d),\leq^{d^{+}}\!)$ . The manner in which this definition is obtained is inspired by Romaguera–Valero theorem and Kostanek–Waszkiewicz theorem. Furthermore, we obtain characterizations of local Yoneda-complete quasi-metric spaces via local nets in quasi-metric spaces. More precisely, we prove that a quasi-metric space is local Yoneda-complete if and only if every local net has a d-limit. Finally, we prove that every quasi-metric space has a local Yoneda completion.

Keywords

Quasi-metric spacelocal Yoneda-complete quasi-metric spaceg-topology
Type
Paper
Information
Mathematical Structures in Computer Science , Volume 33 , Issue 1 , January 2023 , pp. 33 - 45
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Copyright
© The Author(s), 2023. Published by Cambridge University Press

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Footnotes

†

This work is supported by the National Natural Science Foundation of China (Grant Nos. 12101383, 11871320, 11531009) and the Fundamental Research Funds for the Central Universities (Grant No. GK202103006).

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