Answers to Kirk-Shahzad's questions on strong $b$-metric spaces

TL;DR

This paper addresses two open questions about strong $b$-metric spaces, providing a counter-example for one and a completion theorem for the other, advancing understanding of their structure.

Contribution

It constructs a counter-example for an open question and proves a completion theorem, clarifying properties of strong $b$-metric spaces.

Findings

Counter-example negates the first question

Completion theorem affirms the second question

Advances theoretical understanding of strong $b$-metric spaces

Abstract

In this paper, two open questions on strong $b$-metric spaces posed by Kirk and Shahzad [11, Chapter 12] are investigated. A counter-example is constructed to give a negative answer to the first question, and a theorem on the completion of a strong $b$-metric space is proved to give a positive answer to the second question.