Math 2.0

General topology Definition

Topological embedding

Builds on HomeomorphismSubspace topologyInjective functionImage of a set

Definition

Let XX and YY be topological spaces. A function f ⁣:X→Yf \colon X \to Y is a topological embedding when ff is injective and the map X→f(X)X \to f(X), x↦f(x)x \mapsto f(x), is a homeomorphism. Here the image f(X)f(X) has the subspace topology of YY.

Remarks

Each embedding is continuous, because the inclusion of a subspace is continuous. The inclusion of a subspace is an embedding. The map ℝ→ℝ2\mathbb{R} \to \mathbb{R}^2, x↦(x,0)x \mapsto (x, 0), is an embedding. The map t↦(cos⁡t,sin⁡t)t \mapsto (\cos t, \sin t) from [0,2π)[0, 2\pi) to ℝ2\mathbb{R}^2 is continuous and injective, but not an embedding. A continuous injective map from a compact space to a Hausdorff space is an embedding.

Reading path

To understand this page from the ground up, read these 37 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 levelFunctiondefinitionSubspace topology, definition, no evidence levelSubspace topologydefinitionCodomain, definition, no evidence levelCodomaindefinitionInjective function, definition, no evidence levelInjective functiondefinitionImage of a set, definition, no evidence levelImage of a setdefinitionPreimage 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 levelHomeomorphismdefinitionTopological embedding, definition, no evidence levelTopological embeddingdefinition
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. Subspace topology definition
  24. Codomain definition
  25. Injective function definition
  26. Image of a set definition
  27. Preimage of a set definition
  28. Metric space definition
  29. Surjective function definition
  30. Open ball definition
  31. Bijective function definition
  32. Metric topology definition
  33. Inverse function definition
  34. Open set definition
  35. Continuous map of topological spaces definition
  36. Homeomorphism definition
  37. Topological embedding definition

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Embedding, General topology
About this page
Tag08SVRVHJNY
URIm2://08SVRVHJNY
Kinddefinition
WritingW0 stub
MSC54C25
DefinesTopological embedding
UsesHomeomorphism, Subspace topology, Injective function, Image of a set

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