A metric space in which every Cauchy sequence converges to a point of the space (equivalently, every Cauchy net converges when using nets); limits of Cauchy sequences always lie inside the space.

Principle

Completeness ensures that limits of arbitrarily precise approximations exist within the space, so iterative or limit processes do not 'escape' the ambient metric structure.

Demonstration

The real numbers R with the usual metric are complete: every Cauchy sequence of reals converges to a real. The rational numbers Q are not complete because there are Cauchy sequences (e.g. approximations to √2) that have no limit in Q. Closed subsets of complete metric spaces are complete.

Misapplication

Confusing completeness with compactness, or assuming completeness is preserved under arbitrary subspaces (only closed subspaces inherit completeness). Assuming a metric is complete without verifying convergence of Cauchy sequences or completeness of the associated uniformity.

Consequence

Completeness permits application of fixed-point theorems (Banach contraction principle), ensures existence of limits for iterative algorithms, and supports the construction of completions (embedding a space densely into a complete one).

Reversal

An incomplete metric space contains Cauchy sequences that do not converge in the space; completing the space requires adjoining limit points (its completion).

Boundary

Applies to metric (and more generally uniform) spaces; completeness depends on the chosen metric or uniform structure and is not a purely topological property. Finite-dimensional Euclidean spaces are complete; some infinite-dimensional normed spaces may fail completeness.

Semantic Tension

Completeness can be confused with closedness or compactness; closedness is a local/topological notion while completeness is global and metric-dependent, and compactness implies completeness plus total boundedness in metric spaces.

Synthesis

Complete Metric Space = a metric setting in which every sequence of ever closer approximations has its limit inside the space, guaranteeing stability of limit processes and enabling analytic tools that require existence of limits.