Math 2.0

General topology Definition

Path component

Builds on Path-connected spaceEquivalence relation

Definition

Let XX be a topological space. For x,y∈Xx, y \in X, write x≃yx \simeq y when there is a path in XX from xx to yy. This is an equivalence relation on XX. Its classes are the path components of XX.

Remarks

The relation is reflexive by constant paths, symmetric by t↦γ(1−t)t \mapsto \gamma(1 - t), and transitive by joining two paths one after the other. The path component of xx is the largest path-connected subset of XX that contains xx. Each path component lies in a connected component. The topologist's sine curve is one connected component but has two path components. Path components need not be closed.

Reading path

To understand this page from the ground up, read these 34 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 relationdefinitionTopological space, definition, no evidence levelTopological spacedefinitionReflexive relation, definition, no evidence levelReflexive relationdefinitionSymmetric relation, definition, no evidence levelSymmetric relationdefinitionTransitive relation, definition, no evidence levelTransitive relationdefinitionFunction, definition, no evidence levelFunctiondefinitionEquivalence relation, definition, no evidence levelEquivalence relationdefinitionMetric space, definition, no evidence levelMetric spacedefinitionPreimage of a set, definition, no evidence levelPreimage of a setdefinitionOpen ball, definition, no evidence levelOpen balldefinitionMetric topology, definition, no evidence levelMetric topologydefinitionOpen set, definition, no evidence levelOpen setdefinitionContinuous map of topological spaces, definition, no evidence levelContinuous map of top…definitionPath-connected space, definition, no evidence levelPath-connected spacedefinitionPath component, definition, no evidence levelPath componentdefinition
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. Topological space definition
  22. Reflexive relation definition
  23. Symmetric relation definition
  24. Transitive relation definition
  25. Function definition
  26. Equivalence relation definition
  27. Metric space definition
  28. Preimage of a set definition
  29. Open ball definition
  30. Metric topology definition
  31. Open set definition
  32. Continuous map of topological spaces definition
  33. Path-connected space definition
  34. Path component definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Connected_space, Path connectedness
About this page
Tag0SB9FNZECG
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Kinddefinition
WritingW0 stub
MSC54D05
DefinesPath component
UsesPath-connected space, Equivalence relation

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