Math 2.0

General topology Proposition

Connected components are closed

Builds on Connected setClosureUnion of a family of sets

Statement

Let XX be a topological space and let x∈Xx \in X. The connected component of xx is the union C(x)C(x) of all connected subsets of XX that contain xx. Then C(x)C(x) is connected and closed in XX.

Remarks

Proof: a union of connected sets with a common point is connected, so C(x)C(x) is the largest connected set that contains xx. The closure of C(x)C(x) is also connected, so it equals C(x)C(x). The components of XX form a partition of XX. Components need not be open. In ℚ\mathbb{Q}, the components are the single points, which are not open. If XX has only finitely many components, then each of them is also open.

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 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 levelFunctiondefinitionSubspace topology, definition, no evidence levelSubspace topologydefinitionMetric 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 setdefinitionConnected space, definition, no evidence levelConnected spacedefinitionClosure, definition, no evidence levelClosuredefinitionConnected set, definition, no evidence levelConnected setdefinitionConnected components are closed, statement, no evidence levelConnected components …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. Subspace topology definition
  27. Metric space definition
  28. Open ball definition
  29. Metric topology definition
  30. Open set definition
  31. Closed set definition
  32. Connected space definition
  33. Closure definition
  34. Connected set definition
  35. Connected components are closed statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Connected_space, Connected components
About this page
Tag0NRT0E2QT2
URIm2://0NRT0E2QT2
Kindstatement, proposition
WritingW0 stub
MSC54D05
StatusThis copy of the site has no build data. Run m2 status 0NRT0E2QT2.
UsesConnected set, Closure, Union of a family of sets

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