Math 2.0

General topology Connectedness Counterexample

Topologist's sine curve

Builds on Connected setPath-connected spaceSubspace topology

Counterexample

Let T={(x,sin⁡1x):0<x≤1}∪{(0,0)}⊆ℝ2,T = \left\{ \left( x, \sin \tfrac{1}{x} \right) : 0 < x \le 1 \right\} \cup \{ (0, 0) \} \subseteq \mathbb{R}^2, with the subspace topology of the plane. Then TT is connected but not path-connected. Thus connected spaces need not be path-connected.

Remarks

The graph part GG is connected, as the continuous image of (0,1](0, 1]. The origin is in the closure of GG, and a set between a connected set and its closure is connected. Thus TT is connected. No path joins the origin to a point of GG. Near x=0x = 0, the curve goes up and down between −1-1 and 11 infinitely often, and a path cannot follow it continuously. Some texts use the closed version, which adds the whole segment {0}×[−1,1]\{0\} \times [-1, 1]. It is compact, connected and not path-connected. The space TT is also not locally connected.

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 setdefinitionTopologist's sine curve, counterexample, no evidence levelTopologist's sine cur…counterexample
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. Topologist's sine curve counterexample

Further reading

  1. Web https://en.wikipedia.org/wiki/Topologist%27s_sine_curve, lead section
  2. Web https://en.wikipedia.org/wiki/Topologist%27s_sine_curve, Properties
About this page
Tag0K3T4A659R
URIm2://0K3T4A659R
Aliaseswarsaw-sine-curve
Kindcounterexample
WritingW0 stub
MSC54G20
UsesConnected set, Path-connected space, Subspace topology
Same as

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