Math 2.0

General topology Proof

Proof that compact subsets of a metric space are bounded

Builds on Metric spaceCompact setCompactness via open coversOpen ballOpen balls are openBounded setTriangle inequality for a metric

Proof

We cover KK by open balls of growing radius around one point.

  1. Let (X,d)(X, d) be a metric space and let KK be a compact subset of XX.
  2. If K=∅K = \emptyset, then KK is bounded, since the condition on pairs of points is empty. So assume K≠∅K \ne \emptyset and fix a∈Ka \in K.
  3. For each integer n≥1n \ge 1, let B(a,n)B(a, n) be the open ball. It is open, since open balls are open.
  4. Let x∈Kx \in K. Choose an integer n>d(a,x)n > d(a, x). Then x∈B(a,n)x \in B(a, n). So the balls B(a,n)B(a, n) with n≥1n \ge 1 cover KK.
  5. By compactness via open covers, there are integers n1,…,np≥1n_1, \ldots, n_p \ge 1 with K⊆B(a,n1)∪⋯∪B(a,np)K \subseteq B(a, n_1) \cup \cdots \cup B(a, n_p). Since K≠∅K \ne \emptyset, p≥1p \ge 1.
  6. Let N=max⁡(n1,…,np)N = \max(n_1, \ldots, n_p). Each B(a,nj)B(a, n_j) lies in B(a,N)B(a, N), so K⊆B(a,N)K \subseteq B(a, N).
  7. Let x,y∈Kx, y \in K. By the triangle inequality, d(x,y)≤d(x,a)+d(a,y)<2Nd(x, y) \le d(x, a) + d(a, y) < 2N.
  8. So d(x,y)≤2Nd(x, y) \le 2N for all x,y∈Kx, y \in K, and KK is bounded.

Idea

The balls of radius 1,2,3,…1, 2, 3, \ldots around a point exhaust the whole space. Compactness keeps only finitely many of them, and the largest one already holds all of KK.

Reading path

To understand this page from the ground up, read these 40 pages in order. The foundations come first. The path takes the proofs Proof of the description of the open sets of a metric space, Proof that compactness of a subset can be tested with open sets of the ambient space, Proof that open balls are open.

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Element, definition, no evidence levelElementdefinitionProposition, concept, no evidence levelPropositionconceptReal numbers, object, no evidence levelReal numbersobjectSet, definition, no evidence levelSetdefinitionConjunction, definition, no evidence levelConjunctiondefinitionDisjunction, definition, no evidence levelDisjunctiondefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionEmpty set, object, no evidence levelEmpty setobjectFamily of sets, definition, no evidence levelFamily of setsdefinitionOrdered pair, definition, no evidence levelOrdered pairdefinitionIntersection, definition, no evidence levelIntersectiondefinitionUnion, definition, no evidence levelUniondefinitionExistential quantifier, definition, no evidence levelExistential quantifierdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionCartesian product, definition, no evidence levelCartesian productdefinitionSubset, definition, no evidence levelSubsetdefinitionUnion of a family of sets, definition, no evidence levelUnion of a family of …definitionUniqueness quantifier, definition, no evidence levelUniqueness quantifierdefinitionBinary relation, definition, no evidence levelBinary relationdefinitionTopological space, definition, no evidence levelTopological spacedefinitionFunction, definition, no evidence levelFunctiondefinitionSubspace topology, definition, no evidence levelSubspace topologydefinitionMetric space, definition, no evidence levelMetric spacedefinitionClosed ball, definition, no evidence levelClosed balldefinitionOpen ball, definition, no evidence levelOpen balldefinitionTriangle inequality for a metric, statement, no evidence levelTriangle inequality f…statementBounded set, definition, no evidence levelBounded setdefinitionMetric topology, definition, no evidence levelMetric topologydefinitionOpen set, definition, no evidence levelOpen setdefinitionProof of the description of the open sets of a metric space, proof, no evidence levelProof of the descript…proofOpen cover, definition, no evidence levelOpen coverdefinitionOpen sets of a metric space, statement, no evidence levelOpen sets of a metric…statementCompact space, definition, no evidence levelCompact spacedefinitionProof that open balls are open, proof, no evidence levelProof that open balls…proofCompact set, definition, no evidence levelCompact setdefinitionOpen balls are open, statement, no evidence levelOpen balls are openstatementProof that compactness of a subset can be tested with open sets of the ambient space, proof, no evidence levelProof that compactnes…proofCompactness via open covers, statement, no evidence levelCompactness via open …statementProof that compact subsets of a metric space are bounded, proof, no evidence levelProof that compact su…proof
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Element definition
  2. Proposition concept
  3. Real numbers object
  4. Set definition
  5. Conjunction definition
  6. Disjunction definition
  7. Implication definition
  8. Predicate definition
  9. Empty set object
  10. Family of sets definition
  11. Ordered pair definition
  12. Intersection definition
  13. Union definition
  14. Existential quantifier definition
  15. Universal quantifier definition
  16. Cartesian product definition
  17. Subset definition
  18. Union of a family of sets definition
  19. Uniqueness quantifier definition
  20. Binary relation definition
  21. Topological space definition
  22. Function definition
  23. Subspace topology definition
  24. Metric space definition
  25. Closed ball definition
  26. Open ball definition
  27. Triangle inequality for a metric statement
  28. Bounded set definition
  29. Metric topology definition
  30. Open set definition
  31. Proof of the description of the open sets of a metric space proof
  32. Open cover definition
  33. Open sets of a metric space statement
  34. Compact space definition
  35. Proof that open balls are open proof
  36. Compact set definition
  37. Open balls are open statement
  38. Proof that compactness of a subset can be tested with open sets of the ambient space proof
  39. Compactness via open covers statement
  40. Proof that compact subsets of a metric space are bounded proof

Where this leads

703 pages build on this one.

Further reading

  1. Web https://proofwiki.org/wiki/Compact_Subspace_of_Metric_Space_is_Bounded
About this page
TagA81YEQ28RM
URIm2://A81YEQ28RM
Kindproof
WritingW0 stub
MSC54E45
ProvesCompact subsets of a metric space are bounded
UsesMetric space, Compact set, Compactness via open covers, Open ball, Open balls are open, Bounded set, Triangle inequality for a metric
Changed4 November 2026

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