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 $𝑓 : (- 1, 1) \to ℝ : 𝑥 \mapsto 𝑥 (1 - 𝑥 2)$ . While $ℝ$ is complete, $(- 1, 1)$ is not.¹

Any metric space may be embedded in its Metric completion.

Examples

Clearly $ℚ$ with the Euclidean metric is incomplete, whereas $ℝ$ is.
$C 1 [𝑎, 𝑏]$ is not complete, as is showed by a function limiting to absolute value, which is not differentiable at $0$ .

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2020, Topology: A categorical approach, p. 7 ↩

Complete metric space

Examples

Footnotes