Math 2.0

General topology Proposition

Closed sets contain their limits

Builds on Closed setConvergence of a sequence in a metric spaceOpen sets of a metric space

Statement

Let (X,d)(X, d) be a metric space and let FF be a closed subset of XX. Let (xn)(x_n) be a sequence of points of FF that converges to a point x∈Xx \in X. Then x∈Fx \in F.

Remarks

Proof: if x∉Fx \notin F, then X∖FX \setminus F is open and contains xx. Thus some ball B(x,ε)B(x, \varepsilon) misses FF. But xn∈B(x,ε)x_n \in B(x, \varepsilon) for large nn, a contradiction. In a metric space the converse is also true: a set that contains the limits of all its convergent sequences is closed. This follows from closure is the set of limits of sequences. The statement holds in every topological space, with the same proof.

Reading path

To understand this page from the ground up, read these 32 pages in order. The foundations come first.

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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 relationdefinitionUniversal set, definition, no evidence levelUniversal setdefinitionTopological space, definition, no evidence levelTopological spacedefinitionComplement, definition, no evidence levelComplementdefinitionFunction, definition, no evidence levelFunctiondefinitionMetric space, definition, no evidence levelMetric spacedefinitionConvergence of a sequence in a metric space, definition, no evidence levelConvergence of a sequ…definitionOpen ball, definition, no evidence levelOpen balldefinitionMetric topology, definition, no evidence levelMetric topologydefinitionOpen set, definition, no evidence levelOpen setdefinitionOpen sets of a metric space, statement, no evidence levelOpen sets of a metric…statementClosed set, definition, no evidence levelClosed setdefinitionClosed sets contain their limits, statement, no evidence levelClosed sets contain t…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. Universal set definition
  22. Topological space definition
  23. Complement definition
  24. Function definition
  25. Metric space definition
  26. Convergence of a sequence in a metric space definition
  27. Open ball definition
  28. Metric topology definition
  29. Open set definition
  30. Open sets of a metric space statement
  31. Closed set definition
  32. Closed sets contain their limits statement

Where this leads

3 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Closed_set, More about closed sets
About this page
Tag09GQ3ZTQJ9
URIm2://09GQ3ZTQJ9
Kindstatement, proposition
WritingW0 stub
MSC54E35
StatusThis copy of the site has no build data. Run m2 status 09GQ3ZTQJ9.
UsesClosed set, Convergence of a sequence in a metric space, Open sets of a metric space

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