Math 2.0

General topology Proposition

Path-connected spaces are connected

Builds on Path-connected spaceConnected spaceContinuous image of a connected set is connectedConnected subsets of the real line are intervals

Statement

Let XX be a path-connected space. Then XX is connected.

Remarks

Proof: suppose open sets UU and VV split XX, with x∈Ux \in U and y∈Vy \in V. Take a path γ\gamma from xx to yy. Then γ([0,1])\gamma([0, 1]) is connected, by continuous image of a connected set is connected and connectedness of intervals. But UU and VV split it, a contradiction. The converse fails, as the topologist's sine curve shows. It holds for open subsets of R^n.

Reading path

To understand this page from the ground up, read these 50 pages in order. The foundations come first.

  1. Construction of the reals from Cauchy sequences concept
  2. Element definition
  3. Integers object
  4. Ordered field definition
  5. Proposition concept
  6. Rational numbers object
  7. Real numbers object
  8. Set bounded above definition
  9. Set definition
  10. Ring of integers statement
  11. Conjunction definition
  12. Disjunction definition
  13. Implication definition
  14. Predicate definition
  15. Supremum definition
  16. Empty set object
  17. Family of sets definition
  18. Ordered pair definition
  19. Intersection definition
  20. Union definition
  21. The rationals form a field statement
  22. Existential quantifier definition
  23. Universal quantifier definition
  24. Completeness axiom statement
  25. Cartesian product definition
  26. Subset definition
  27. Union of a family of sets definition
  28. The rationals form an ordered field statement
  29. Uniqueness quantifier definition
  30. Binary relation definition
  31. Universal set definition
  32. Order on the reals definition
  33. Topological space definition
  34. Complement definition
  35. Function definition
  36. Interval definition
  37. Subspace topology definition
  38. Preimage of a set definition
  39. Metric space definition
  40. Open ball definition
  41. Metric topology definition
  42. Open set definition
  43. Closed set definition
  44. Continuous map of topological spaces definition
  45. Connected space definition
  46. Path-connected space definition
  47. Connected set definition
  48. Continuous image of a connected set is connected statement
  49. Connected subsets of the real line are intervals statement
  50. Path-connected spaces are connected statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Connected_space, Path connectedness
About this page
Tag0ZMT1GP4KM
URIm2://0ZMT1GP4KM
Kindstatement, proposition
WritingW0 stub
MSC54D05
StatusThis copy of the site has no build data. Run m2 status 0ZMT1GP4KM.
UsesPath-connected space, Connected space, Continuous image of a connected set is connected, Connected subsets of the real line are intervals

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