1. Conversely, on a uniform space every convergent filter is a Cauchy filter . 2. Moreover, every cluster point of a Cauchy filter is a limit point. 3. Even more generally, Cauchy spaces are spaces in which Cauchy filters may be defined. 4. Every Cauchy sequence is bounded, although Cauchy nets or Cauchy filters may not be bounded. 5. Instead of working with Cauchy sequences, one works with Cauchy filters ( or Cauchy nets ). 6. Conversely, a uniform space is called "'complete "'if every Cauchy filter converges. 7. It is also possible to replace Cauchy " sequences " in the definition of completeness by Cauchy " Cauchy filters . 8. In a metric space this agrees with the previous definition . " X " is said to be complete if every Cauchy filter converges. 9. If every Cauchy net ( or equivalently every Cauchy filter ) has a limit in " X ", then " X " is called complete. 10. More generally, in a Cauchy space, a net ( " x " ? ) is Cauchy if the filter generated by the net is a Cauchy filter .
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