Mathematics.

point set topology

Compactness in Metric Spaces

Topology45 minDifficulty8 out of 10

You should know: compactness, metric spaces

Overview

In a general topological space, compactness is defined via open covers, but in a metric space this abstract condition turns out to be equivalent to two more concrete, hands-on properties: sequential compactness (every sequence has a subsequence converging to a point of the space) and the combination of completeness and total boundedness. This three-way equivalence — compactness ⟺ sequential compactness ⟺ complete + totally bounded — is one of the central theorems of metric space theory, and it specializes further in Euclidean space ℝⁿ to the Heine–Borel theorem: compact ⟺ closed and bounded. The equivalence lets analysts freely switch between the cover-based definition (good for proving abstract theorems) and the sequence-based definition (good for concrete computations) whenever a metric is present.

Intuition

Think of trying to trap a set with a finite number of small nets (open cover compactness) versus trying to escape to infinity or squeeze into an ever-thinner gap along a sequence (sequential compactness). In a metric space these turn out to be the same struggle: if you can always extract a convergent subsequence from any infinite sequence of points, you can also always finitely cover the space, and vice versa — because the metric lets you convert 'no finite subcover' into 'a sequence with no convergent subsequence' by picking one point from each leftover, uncovered region at every stage. Total boundedness sharpens 'bounded' to mean the set can be covered by finitely many balls of any prescribed tiny radius (not just one huge ball), while completeness guarantees no sequence can 'try to converge' to a point missing from the space — together, these two down-to-earth conditions turn out to force compactness.

Formal Definition

Definition

For a metric space (X,d), the following are equivalent for a subset K ⊆ X:

(i) K is compact (every open cover has a finite subcover)\text{(i) } K \text{ is compact (every open cover has a finite subcover)}
Open-cover compactness
(ii) K is sequentially compact: every sequence in K has a subsequence converging to a point of K\text{(ii) } K \text{ is sequentially compact: every sequence in } K \text{ has a subsequence converging to a point of } K
Sequential compactness
(iii) K is complete and totally bounded: ∀ε>0, K is covered by finitely many balls of radius ε\text{(iii) } K \text{ is complete and totally bounded: } \forall \varepsilon>0,\ K \text{ is covered by finitely many balls of radius } \varepsilon
Complete + totally bounded
(Heine–Borel) K⊆Rn is compact  ⟺  K is closed and bounded\text{(Heine–Borel) } K \subseteq \mathbb{R}^n \text{ is compact} \iff K \text{ is closed and bounded}

A special case of the equivalence above: in ℝⁿ, total boundedness is equivalent to plain boundedness, and completeness of a subset reduces to being closed

Heine–Borel theorem

Notation

NotationMeaning
∀ε>0, ∃x1,…,xn∈K:K⊆⋃i=1nB(xi,ε)\forall \varepsilon>0,\ \exists x_1,\ldots,x_n \in K: K \subseteq \bigcup_{i=1}^n B(x_i,\varepsilon)K can be covered by finitely many balls of any given radius ε
every Cauchy sequence in K converges to a point of K\text{every Cauchy sequence in } K \text{ converges to a point of } KK has no 'missing limit points' that a Cauchy sequence could be trying to approach

Properties

Compact ⟹ sequentially compact (general topology)

In any metric space, compactness and sequential compactness coincide.\text{In any metric space, compactness and sequential compactness coincide.}

Condition: This equivalence can fail in general (non-metrizable) topological spaces, where the two notions can diverge.

Total boundedness is strictly stronger than boundedness

Totally bounded  ⟹  bounded, but not conversely in general metric spaces\text{Totally bounded} \implies \text{bounded}, \text{ but not conversely in general metric spaces}

Example: The closed unit ball in an infinite-dimensional Hilbert space is bounded but not totally bounded, hence not compact.

Heine–Borel is special to finite dimensions

In Rn, bounded  ⟺  totally bounded, so compact  ⟺  closed and bounded.\text{In } \mathbb{R}^n, \text{ bounded} \iff \text{totally bounded}, \text{ so compact} \iff \text{closed and bounded.}

Condition: This finite-dimensional collapse of 'bounded' into 'totally bounded' fails in infinite-dimensional normed spaces.

Worked Examples

  1. Split into even and odd subsequences: x_{2k} = 1 - 1/(2k) → 1, and x_{2k+1} = -(1-1/(2k+1)) → -1.

    x2k→1,x2k+1→−1x_{2k} \to 1, \quad x_{2k+1} \to -1

  2. Since xₙ ∈ [0,1] must actually hold for the example to apply, restrict to the even subsequence, which converges to 1 ∈ [0,1].

    x2k=1−12k→1∈[0,1]x_{2k} = 1 - \tfrac{1}{2k} \to 1 \in [0,1]

Answer: The subsequence x_{2k} converges to 1, a point of [0,1], illustrating sequential compactness.

Practice Problems

Difficulty 6/10

In a metric space, compactness is equivalent to:

Difficulty 7/10

Is the set of rationals in [0,1] (with the usual metric) compact?

Difficulty 8/10

Using Heine–Borel, prove that the closed unit ball B = {x ∈ ℝⁿ : ‖x‖ ≤ 1} is compact.

Quiz

Sequential compactness in a metric space means:
By the Heine–Borel theorem, compactness in ℝⁿ is equivalent to being:

Summary

  • In a metric space, compactness (open-cover), sequential compactness (convergent subsequences), and completeness + total boundedness are all equivalent conditions.
  • Total boundedness (coverable by finitely many balls of any radius) is strictly stronger than mere boundedness in general metric spaces, though the two coincide in ℝⁿ.
  • The Heine–Borel theorem is the ℝⁿ special case: compact ⟺ closed and bounded, since boundedness and total boundedness agree in finite dimensions.

References

  1. WebsiteWikipedia — Compact space