Math 2.0

General topology Definition

Homeomorphism

Builds on Continuous map of topological spacesBijective functionInverse function

Definition

Let XX and YY be topological spaces. A function f ⁣:X→Yf \colon X \to Y is a homeomorphism when ff is bijective, ff is continuous, and the inverse function f−1 ⁣:Y→Xf^{-1} \colon Y \to X is continuous. The spaces XX and YY are homeomorphic when there is a homeomorphism X→YX \to Y.

Remarks

Equivalently, ff is a bijection and, for each U⊆XU \subseteq X, the set UU is open if and only if f(U)f(U) is open. The map x↦x/(1−∣x∣)x \mapsto x / (1 - |x|) is a homeomorphism from (−1,1)(-1, 1) to ℝ\mathbb{R}. A continuous bijection need not be a homeomorphism: t↦(cos⁡t,sin⁡t)t \mapsto (\cos t, \sin t) from [0,2π)[0, 2\pi) to the circle is one example. For a compact domain and a Hausdorff target, it is always a homeomorphism. Homeomorphisms keep each topological property.

Reading path

To understand this page from the ground up, read these 34 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 relationdefinitionTopological space, definition, no evidence levelTopological spacedefinitionFunction, definition, no evidence levelFunctiondefinitionCodomain, definition, no evidence levelCodomaindefinitionInjective function, definition, no evidence levelInjective functiondefinitionPreimage of a set, definition, no evidence levelPreimage of a setdefinitionMetric space, definition, no evidence levelMetric spacedefinitionSurjective function, definition, no evidence levelSurjective functiondefinitionOpen ball, definition, no evidence levelOpen balldefinitionBijective function, definition, no evidence levelBijective functiondefinitionMetric topology, definition, no evidence levelMetric topologydefinitionInverse function, definition, no evidence levelInverse functiondefinitionOpen set, definition, no evidence levelOpen setdefinitionContinuous map of topological spaces, definition, no evidence levelContinuous map of top…definitionHomeomorphism, definition, no evidence levelHomeomorphismdefinition
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. Topological space definition
  22. Function definition
  23. Codomain definition
  24. Injective function definition
  25. Preimage of a set definition
  26. Metric space definition
  27. Surjective function definition
  28. Open ball definition
  29. Bijective function definition
  30. Metric topology definition
  31. Inverse function definition
  32. Open set definition
  33. Continuous map of topological spaces definition
  34. Homeomorphism definition

Where this leads

33 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Homeomorphism, Definition
About this page
Tag05GZ11PWRY
URIm2://05GZ11PWRY
Kinddefinition
WritingW1 stated
MSC54C05
DefinesHomeomorphism
UsesContinuous map of topological spaces, Bijective function, Inverse function

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