Mathematics.

metric space analysis

Completeness of Metric Spaces

Real Analysis40 minDifficulty5 out of 10

You should know: metric spaces, cauchy sequences

Overview

A metric space (X, d) is complete if every Cauchy sequence in X converges to a limit that lies in X itself. This generalizes the completeness of ℝ (with the usual absolute-value metric) to arbitrary spaces equipped with a notion of distance: ℝⁿ with the Euclidean metric is complete, the space C[a,b] of continuous functions with the sup-norm metric is complete, but ℚ (with the usual metric) is not, since Cauchy sequences of rationals can converge to irrational 'holes' outside ℚ. Completeness is not a purely topological property (two metrics can induce the same topology while only one is complete), but it is preserved by uniform equivalence of metrics, and every metric space embeds isometrically into a complete one, its completion — exactly how ℝ itself is constructed as the completion of ℚ.

Intuition

Completeness of a metric space says the space has 'no missing points that Cauchy sequences are trying to reach.' If you build a sequence whose terms keep getting closer and closer to each other (Cauchy), completeness guarantees there's actually a point in the space that they're converging to — nothing falls through a crack. ℚ has cracks at every irrational number: you can build rational sequences that huddle together tighter and tighter, aiming at √2, but √2 itself isn't in ℚ, so the sequence has nowhere to land within the space. The completion process patches every such crack by literally adjoining the missing limit points — this is one modern way to construct ℝ from ℚ, and the same abstract process builds the completion of any metric space.

Formal Definition

Definition

A metric space (X, d) is complete if:

∀(xn)⊆X Cauchy (i.e. ∀ε>0 ∃N:∀m,n≥N, d(xm,xn)<ε), ∃x∈X:xn→x\forall (x_n) \subseteq X \text{ Cauchy (i.e. } \forall \varepsilon>0\ \exists N: \forall m,n\ge N,\ d(x_m,x_n)<\varepsilon\text{)},\ \exists x \in X: x_n \to x
Definition of a complete metric space
(Rn,∥⋅∥2) is complete;(Q,∣⋅∣) is not complete(\mathbb{R}^n, \|\cdot\|_2) \text{ is complete}; \quad (\mathbb{Q}, |\cdot|) \text{ is not complete}
Canonical examples
Every metric space (X,d) embeds isometrically as a dense subset of a complete metric space (X^,d^) (its completion)\text{Every metric space } (X,d) \text{ embeds isometrically as a dense subset of a complete metric space } (\hat{X}, \hat{d}) \text{ (its completion)}
Completion theorem

Worked Examples

  1. Consider aₙ = 1/n for n ≥ 2, so the sequence lies in (0,1) and is Cauchy (as shown for 1/n in ℝ generally).

    an=1n∈(0,1) for n≥2a_n = \frac{1}{n} \in (0,1) \text{ for } n \ge 2

  2. In ℝ, aₙ → 0, but 0 ∉ (0,1), so within the space (0,1) this Cauchy sequence has no limit.

    an→0∉(0,1)a_n \to 0 \notin (0,1)

Answer: (0,1) is not complete: the Cauchy sequence 1/n has no limit inside (0,1) itself.

Practice Problems

Difficulty 5/10

Is ℤ (the integers) with the usual metric d(m,n)=|m-n| a complete metric space? Justify briefly.

Difficulty 4/10

Which of these metric spaces is NOT complete?

Difficulty 6/10

Prove that a closed subset F of a complete metric space (X,d) is itself complete (with the restricted metric).

Quiz

A metric space (X,d) is complete if:
Which classical construction of ℝ exploits the idea of completion of a metric space?

Summary

  • A metric space is complete if every Cauchy sequence within it converges to a point of the space itself.
  • ℝⁿ and C[a,b] (sup-norm) are complete; ℚ and open subsets like (0,1) are generally not.
  • Every metric space embeds isometrically into a complete space, its completion — the same construction that builds ℝ from ℚ.

References

  1. WebsiteWikipedia — Complete metric space