Math 2.0

General topology Definition

Boundary

Builds on ClosureInteriorSet difference

Definition

Let XX be a topological space and let S⊆XS \subseteq X. The boundary of SS is the set difference ∂S=S‾∖int⁡S,\partial S = \overline{S} \setminus \operatorname{int} S, where S‾\overline{S} is the closure and int⁡S\operatorname{int} S is the interior of SS.

Remarks

A point xx is in ∂S\partial S if and only if each open set that contains xx meets both SS and X∖SX \setminus S. The boundary is closed, and ∂S=∂(X∖S)\partial S = \partial (X \setminus S). In ℝ\mathbb{R}, the boundary of (0,1)(0, 1) is {0,1}\{0, 1\}, and the boundary of ℚ\mathbb{Q} is ℝ\mathbb{R}. Another name for the boundary is the frontier.

Reading path

To understand this page from the ground up, read these 35 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 levelImplicationdefinitionNegation, definition, no evidence levelNegationdefinitionPredicate, 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 levelUniondefinitionSet difference, definition, no evidence levelSet differencedefinitionExistential 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 setdefinitionClosed set, definition, no evidence levelClosed setdefinitionInterior, definition, no evidence levelInteriordefinitionClosure, definition, no evidence levelClosuredefinitionBoundary, definition, no evidence levelBoundarydefinition
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. Negation definition
  9. Predicate definition
  10. Empty set object
  11. Family of sets definition
  12. Ordered pair definition
  13. Intersection definition
  14. Union definition
  15. Set difference definition
  16. Existential quantifier definition
  17. Universal quantifier definition
  18. Cartesian product definition
  19. Subset definition
  20. Intersection of a family of sets definition
  21. Union of a family of sets definition
  22. Uniqueness quantifier definition
  23. Binary relation definition
  24. Universal set definition
  25. Topological space definition
  26. Complement definition
  27. Function definition
  28. Metric space definition
  29. Open ball definition
  30. Metric topology definition
  31. Open set definition
  32. Closed set definition
  33. Interior definition
  34. Closure definition
  35. Boundary definition

Where this leads

5 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Boundary_(topology), Definitions
About this page
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WritingW1 stated
MSC54A05
DefinesBoundary
UsesClosure, Interior, Set difference

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