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

## TL;DR

This paper affirms that Cantor's intersection theorem extends to $\

## Contribution

It proves the extension of Cantor's intersection theorem to $\\mathcal{F}$-metric spaces and shows equivalence of compactness notions with usual metric spaces.

## Key findings

- Cantor's intersection theorem holds in $\\mathcal{F}$-metric spaces.
- Compactness and total boundedness are equivalent to those in metric spaces.
- Provides an affirmative answer to an open problem in generalized 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.