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.