Math 2.0

General topology Proof

Proof that the interior is open

Builds on Topological spaceInteriorOpen setUnion of a family of sets

Proof

  1. Let (X,τ)(X, \tau) be a topological space and let S⊆XS \subseteq X.
  2. Let 𝒰\mathcal{U} be the family of all open sets UU of XX with U⊆SU \subseteq S. Each member of 𝒰\mathcal{U} is in τ\tau.
  3. By the definition of the interior, int⁡S\operatorname{int} S is the union of the members of 𝒰\mathcal{U}.
  4. By the second axiom of a topology, the union of each family of members of τ\tau is in τ\tau.
  5. So int⁡S∈τ\operatorname{int} S \in \tau, that is, int⁡S\operatorname{int} S is open in XX.

Idea

The interior is built as a union of open sets. The axioms of a topology say that every such union is open. So no work is needed beyond the definition.

Reading path

To understand this page from the ground up, read these 28 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 levelFunctiondefinitionMetric space, definition, no evidence levelMetric spacedefinitionOpen ball, definition, no evidence levelOpen balldefinitionMetric topology, definition, no evidence levelMetric topologydefinitionOpen set, definition, no evidence levelOpen setdefinitionInterior, definition, no evidence levelInteriordefinitionProof that the interior is open, proof, no evidence levelProof that the interi…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.
  • 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. Metric space definition
  24. Open ball definition
  25. Metric topology definition
  26. Open set definition
  27. Interior definition
  28. Proof that the interior is open proof

Where this leads

8 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Interior_(topology)
About this page
TagJDTQVBV3CE
URIm2://JDTQVBV3CE
Kindproof
WritingW0 stub
MSC54A05
ProvesInterior is open
UsesTopological space, Interior, Open set, Union of a family of sets
Changed4 November 2026

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