Complete metric space

A complete metric space is a metric space for which every Cauchy sequence is a convergent sequence, i.e. the limit of a sequence is in the space. A stronger condition is compactness.

Completeness is not a topological property

Completeness is not a topological property, unlike the stronger sequential compactness. Consider the homeomorphism . While is complete, is not.¹

Any metric space may be embedded in its Metric completion.

Examples

Clearly with the Euclidean metric is incomplete, whereas is.
is not complete, as is showed by a function limiting to absolute value, which is not differentiable at .

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