A note on I-convergence in quasi-metric spaces

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.