Cantor's intersection theorem in the setting of $\mathcal{F}$-metric spaces

Abstract

This paper deals with an open problem posed by Jleli and Samet in \cite[\, M.~Jleli and B.~Samet, On a new generalization of metric spaces, J. Fixed Point Theory Appl, 20(3) 2018]{JS1}. In \cite[\, Remark 5.1]{JS1} They asked whether the Cantor's intersection theorem can be extended to $\mathcal{F}$-metric spaces or not. In this manuscript we give an affirmative answer to this open question. We also show that the notions of compactness, totally boundedness in the setting of $\mathcal{F}$-metric spaces are equivalent to that of usual metric spaces.