Math 2.0

General topology Proposition

Closure is the set of limits of sequences

Builds on Closure in a metric spaceConvergence of a sequence in a metric spaceClosed sets contain their limits

Statement

Let (X,d)(X, d) be a metric space, let S⊆XS \subseteq X and let x∈Xx \in X. Then xx is in the closure S‾\overline{S} if and only if there is a sequence (sn)(s_n) of points of SS that converges to xx.

Remarks

Proof of the direct part: by closure in a metric space, for each n≥1n \ge 1 there is sn∈Ss_n \in S with d(x,sn)<1/nd(x, s_n) < 1/n. Then sn→xs_n \to x. The converse follows because S‾\overline{S} is closed and closed sets contain their limits. A consequence: SS is closed if and only if the limit of each convergent sequence of points of SS is in SS. In a general topological space, sequences are not enough, and one uses nets or filters instead.

Reading path

To understand this page from the ground up, read these 36 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 levelSubsetdefinitionIntersection of a family of sets, definition, no evidence levelIntersection of a fam…definitionUnion 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…statementClosure, definition, no evidence levelClosuredefinitionClosure in a metric space, statement, no evidence levelClosure in a metric s…statementClosure is the set of limits of sequences, statement, no evidence levelClosure is the set of…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.
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  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. Intersection of a family of sets definition
  19. Union of a family of sets definition
  20. Uniqueness quantifier definition
  21. Binary relation definition
  22. Universal set definition
  23. Topological space definition
  24. Complement definition
  25. Function definition
  26. Metric space definition
  27. Convergence of a sequence in a metric space definition
  28. Open ball definition
  29. Metric topology definition
  30. Open set definition
  31. Open sets of a metric space statement
  32. Closed set definition
  33. Closed sets contain their limits statement
  34. Closure definition
  35. Closure in a metric space statement
  36. Closure is the set of limits of sequences statement

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Closure_(topology), Point of closure
About this page
Tag003QP7A8HG
URIm2://003QP7A8HG
Kindstatement, proposition
WritingW0 stub
MSC54E35
StatusThis copy of the site has no build data. Run m2 status 003QP7A8HG.
UsesClosure in a metric space, Convergence of a sequence in a metric space, Closed sets contain their limits

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