In general a field without this property can be extended into with it by finding the set of equivalence classes of all the Cauchy sequences, as seen by the construction of the reals from the rationals.
For a space with this property, the intersection of a countable number of open dense sets is also dense, as shown in the Baire category theorem. All closed and bounded sets are compact only if the space holds this property, and spaces with this property are compact if and only if they are also totally bounded. Hilbert space is the extension of an inner product space to make it have this property. In general a field without this property can be extended into with it by finding the set of equivalence classes of all the Cauchy sequences, as seen by the construction of the reals from the rationals. For 10 points, name this property of a metric space in which every Cauchy sequence converges to a point in the space.
For a space with this property, the intersection of a countable number of open dense sets is also dense, as shown in the Baire category theorem.
Cauchy completeness [accept just "Cauchy" before mentioned]