A note on $\mathcal{I}$-convergence in quasi-metric spaces

Abstract

In this paper, we define several ideal versions of Cauchy sequences and completeness in quasi-metric spaces. Some examples are constructed to clarify their relationships. We also show that: (1) if a quasi-metric space $(X,\rho)$ is $\mathcal{I}$-sequentially complete, for each decreasing sequence $\{F_n\}$ of nonempty $\mathcal{I}$-closed sets with diam$\{F_n\}\rightarrow 0$ as $n\rightarrow \infty$, then $\bigcap_{n\in \mathbb N}F_n$ is a single-point set; (2) let $\mathcal{I}$ be a $P$-ideal, then every precompact left $\mathcal{I}$-sequentially complete quasi-metric space is compact.