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