Math 2.0

General topology Proposition

Compact subsets of a metric space are bounded

Builds on Compactness via open coversBounded setOpen ball

Statement

Let (X,d)(X, d) be a metric space and let KK be a compact subset of XX. Then KK is bounded.

Remarks

Proof: if KK is empty, it is bounded. Otherwise fix x∈Kx \in K. The open balls B(x,n)B(x, n) for n≥1n \ge 1 cover KK, so finitely many of them do. The largest of these balls contains KK. A compact subset of a metric space is also closed, because metric spaces are Hausdorff. The converse fails in general: a closed bounded set need not be compact. With the metric min⁡(∣x−y∣,1)\min(|x - y|, 1), the real line is closed and bounded, but not compact.

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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 balldefinitionBounded set, definition, no evidence levelBounded setdefinitionMetric topology, definition, no evidence levelMetric topologydefinitionOpen set, definition, no evidence levelOpen setdefinitionOpen cover, definition, no evidence levelOpen coverdefinitionCompact space, definition, no evidence levelCompact spacedefinitionCompact set, definition, no evidence levelCompact setdefinitionCompactness via open covers, statement, no evidence levelCompactness via open …statementCompact subsets of a metric space are bounded, statement, no evidence levelCompact subsets of a …statement
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. Bounded set definition
  28. Metric topology definition
  29. Open set definition
  30. Open cover definition
  31. Compact space definition
  32. Compact set definition
  33. Compactness via open covers statement
  34. Compact subsets of a metric space are bounded statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Compact_space, Metric spaces
About this page
Tag0R7BTEWCJ2
URIm2://0R7BTEWCJ2
Kindstatement, proposition
WritingW0 stub
MSC54E45
StatusThis copy of the site has no build data. Run m2 status 0R7BTEWCJ2.
UsesCompactness via open covers, Bounded set, Open ball

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