Math 2.0

General topology Proof

Proof that the inclusion of a subspace is continuous

Builds on Subspace topologyContinuous map of topological spacesPreimage of a set

Proof

We compute preimages of open sets.

  1. Let UU be open in XX. The preimage is ι−1(U)={s∈S:ι(s)∈U}\iota^{-1}(U) = \set{s \in S : \iota(s) \in U}.
  2. Since ι(s)=s\iota(s) = s, this is {s∈S:s∈U}=S∩U\set{s \in S : s \in U} = S \cap U.
  3. By the definition of the subspace topology, S∩US \cap U is open in SS.
  4. So the preimage of each open set of XX is open in SS, and ι\iota is continuous.

Idea

The subspace topology is built from the sets S∩US \cap U with UU open in XX. These are exactly the preimages of open sets under the inclusion, so the inclusion is continuous by design.

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 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…definitionProof that the inclusion of a subspace is continuous, proof, no evidence levelProof that the inclus…proof
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. Proof that the inclusion of a subspace is continuous proof

Where this leads

4 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Subspace_topology, Properties
About this page
Tag8ZHBEVTFZ9
URIm2://8ZHBEVTFZ9
Kindproof
WritingW0 stub
MSC54B05
ProvesInclusion of a subspace is continuous
UsesSubspace topology, Continuous map of topological spaces, Preimage of a set
Changed4 November 2026

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