A note on I-convergence in quasi-metric spaces

Abstract

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