Question

4. (a) Define a complete metric space.

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Answer 1

Answer:

In mathematical analysis, a metric space M is called complete if every Cauchy sequence of points in M has a limit that is also in M or, alternatively, if every Cauchy sequence in M converges in M. Intuitively, a space is complete if there are no "points missing" from it.

Answer 2

Answer:

Definition: A metric space (X, d) is complete if any of the following equivalent conditions are satisfied:

Every Cauchy sequence of points in M has a limit that is also in M.

Every Cauchy sequence in M converges in M (to some points of M).

The expansion constant of (X, d) is ≤ 2.