Math 2.0

General topology Proposition

Inclusion of a subspace is continuous

Builds on Subspace topologyContinuous map of topological spacesPreimage of a set

Statement

Let XX be a topological space, let S⊆XS \subseteq X have the subspace topology and let ι ⁣:S→X\iota \colon S \to X be the inclusion map ι(s)=s\iota(s) = s. Then ι\iota is continuous.

Remarks

Proof: for UU open in XX, the preimage is ι−1(U)=S∩U\iota^{-1}(U) = S \cap U, which is open in SS. Thus the restriction f∣S=f∘ιf|_S = f \circ \iota of a continuous map f ⁣:X→Yf \colon X \to Y is continuous. A map g ⁣:Z→Sg \colon Z \to S is continuous if and only if ι∘g ⁣:Z→X\iota \circ g \colon Z \to X is continuous. This is the universal property of the subspace topology.

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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 topologydefinitionPreimage of a set, definition, no evidence levelPreimage of a setdefinitionMetric space, definition, no evidence levelMetric spacedefinitionOpen ball, definition, no evidence levelOpen balldefinitionMetric topology, definition, no evidence levelMetric topologydefinitionOpen set, definition, no evidence levelOpen setdefinitionContinuous map of topological spaces, definition, no evidence levelContinuous map of top…definitionInclusion of a subspace is continuous, statement, no evidence levelInclusion of a subspa…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. Topological space definition
  22. Function definition
  23. Subspace topology definition
  24. Preimage of a set definition
  25. Metric space definition
  26. Open ball definition
  27. Metric topology definition
  28. Open set definition
  29. Continuous map of topological spaces definition
  30. Inclusion of a subspace is continuous statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Subspace_topology, Properties
About this page
Tag065J822CKN
URIm2://065J822CKN
Kindstatement, proposition
WritingW0 stub
MSC54B05
StatusThis copy of the site has no build data. Run m2 status 065J822CKN.
UsesSubspace topology, Continuous map of topological spaces, Preimage of a set

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