Math 2.0

General topology Proposition

Continuous image of a connected set is connected

Builds on Connected setContinuous map of topological spacesPreimage of a set

Statement

Let XX and YY be topological spaces, let f ⁣:X→Yf \colon X \to Y be continuous and let S⊆XS \subseteq X be connected. Then f(S)f(S) is connected.

Remarks

Proof: if open sets UU and VV split f(S)f(S), then the preimages f−1(U)f^{-1}(U) and f−1(V)f^{-1}(V) are open and split SS. Thus connectedness is a topological property. For Y=ℝY = \mathbb{R}, this gives the intermediate value theorem. For example, the circle is connected, because it is the image of [0,1][0, 1] under t↦(cos⁡2πt,sin⁡2πt)t \mapsto (\cos 2\pi t, \sin 2\pi t).

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 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 spacedefinitionConnected set, definition, no evidence levelConnected setdefinitionContinuous image of a connected set is connected, statement, no evidence levelContinuous image of a…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.
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  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. Connected set definition
  35. Continuous image of a connected set is connected statement

Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Connected_space, Theorems
About this page
Tag0EPD87YVG8
URIm2://0EPD87YVG8
Kindstatement, proposition
WritingW0 stub
MSC54D05
StatusThis copy of the site has no build data. Run m2 status 0EPD87YVG8.
UsesConnected set, Continuous map of topological spaces, Preimage of a set

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