Math 2.0

General topology Theorem

Open connected subsets of R^n are path-connected

Builds on Connected setPath-connected spaceOpen ball

Statement

Let n≥1n \ge 1 and let U⊆ℝnU \subseteq \mathbb{R}^n be open and connected, for the Euclidean topology. Then UU is path-connected or empty.

Remarks

Proof: fix x∈Ux \in U, and let PP be the set of points of UU that a path in UU joins to xx. Each open ball in UU is convex, so PP and U∖PU \setminus P are both open. As UU is connected and x∈Px \in P, we get P=UP = U. The proof works in each locally path-connected space, for example in each open subset of a normed space. In fact, any two points of UU can be joined by a path made of finitely many line segments.

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 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 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 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 setdefinitionClosed set, definition, no evidence levelClosed setdefinitionContinuous map of topological spaces, definition, no evidence levelContinuous map of top…definitionConnected space, definition, no evidence levelConnected spacedefinitionPath-connected space, definition, no evidence levelPath-connected spacedefinitionConnected set, definition, no evidence levelConnected setdefinitionOpen connected subsets of R^n are path-connected, statement, no evidence levelOpen connected subset…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. Union of a family of sets definition
  19. Uniqueness quantifier definition
  20. Binary relation definition
  21. Universal set definition
  22. Topological space definition
  23. Complement definition
  24. Function definition
  25. Subspace topology definition
  26. Metric space definition
  27. Preimage of a set definition
  28. Open ball definition
  29. Metric topology definition
  30. Open set definition
  31. Closed set definition
  32. Continuous map of topological spaces definition
  33. Connected space definition
  34. Path-connected space definition
  35. Connected set definition
  36. Open connected subsets of R^n are path-connected statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Connected_space, Path connectedness
About this page
Tag0MXN7AFBPE
URIm2://0MXN7AFBPE
Kindstatement, theorem
WritingW0 stub
MSC54D05
StatusThis copy of the site has no build data. Run m2 status 0MXN7AFBPE.
UsesConnected set, Path-connected space, Open ball

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