Math 2.0

General topology Definition

Connected space

Builds on Topological spaceOpen setClosed set

Definition

A topological space XX is connected when there are no open sets U,V⊆XU, V \subseteq X with U≠∅U \neq \varnothing, V≠∅V \neq \varnothing, U∩V=∅U \cap V = \varnothing and U∪V=XU \cup V = X.

Remarks

Equivalently, the only subsets of XX that are both open and closed are ∅\varnothing and XX. Equivalently, each continuous map from XX to the discrete space {0,1}\{0, 1\} is constant. The interval [0,1][0, 1] is connected. The set [0,1]∪[2,3][0, 1] \cup [2, 3] is not connected. Under this definition the empty space is connected. Some authors require a connected space to be nonempty.

Reading path

To understand this page from the ground up, read these 30 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 levelFunctiondefinitionMetric space, definition, no evidence levelMetric spacedefinitionOpen ball, definition, no evidence levelOpen balldefinitionMetric topology, definition, no evidence levelMetric topologydefinitionOpen set, definition, no evidence levelOpen setdefinitionClosed set, definition, no evidence levelClosed setdefinitionConnected space, definition, no evidence levelConnected spacedefinition
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. Metric space definition
  26. Open ball definition
  27. Metric topology definition
  28. Open set definition
  29. Closed set definition
  30. Connected space definition

Where this leads

14 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Connected_space, Definition
About this page
Tag0D939CMME7
URIm2://0D939CMME7
Kinddefinition
WritingW1 stated
MSC54D05
DefinesConnected space
UsesTopological space, Open set, Closed set

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