Math 2.0

General topology Proposition

Closure in a metric space

Builds on ClosureOpen ballOpen sets of a metric space

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 for each real ε>0\varepsilon > 0 there is s∈Ss \in S with d(x,s)<εd(x, s) < \varepsilon.

Remarks

Equivalently, x∈S‾x \in \overline{S} if and only if each open ball B(x,ε)B(x, \varepsilon) meets SS. For nonempty SS, this means d(x,S)=0d(x, S) = 0, where d(x,S)d(x, S) is the distance from the point to the set. In ℝ\mathbb{R}, the point 00 is in the closure of {1/n:n≥1}\{1/n : n \ge 1\}, because 1/n<ε1/n < \varepsilon for large nn.

Reading path

To understand this page from the ground up, read these 33 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 spacedefinitionOpen 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 setdefinitionClosure, definition, no evidence levelClosuredefinitionClosure in a metric space, statement, no evidence levelClosure in a metric s…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. 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. Open ball definition
  28. Metric topology definition
  29. Open set definition
  30. Open sets of a metric space statement
  31. Closed set definition
  32. Closure definition
  33. Closure in a metric space statement

Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Closure_(topology), Definitions
About this page
Tag068P0P32DW
URIm2://068P0P32DW
Kindstatement, proposition
WritingW0 stub
MSC54E35
StatusThis copy of the site has no build data. Run m2 status 068P0P32DW.
UsesClosure, Open ball, Open sets of a metric space

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