Math 2.0

General topology Proof

Proof that the interior is the largest open subset

Builds on Topological spaceInteriorOpen setUnion of a family of setsInterior is open

Proof

  1. Let XX be a topological space and let SβŠ†XS \subseteq X.
  2. Let 𝒰\mathcal{U} be the family of all open sets UU of XX with UβŠ†SU \subseteq S. By the definition of the interior, int⁑S\operatorname{int} S is the union of the members of 𝒰\mathcal{U}.
  3. Let x∈int⁑Sx \in \operatorname{int} S. Then x∈Ux \in U for some Uβˆˆπ’°U \in \mathcal{U}. Since UβŠ†SU \subseteq S, we get x∈Sx \in S. So int⁑SβŠ†S\operatorname{int} S \subseteq S.
  4. By interior is open, the set int⁑S\operatorname{int} S is open.
  5. Let UU be an open set of XX with UβŠ†SU \subseteq S. Then Uβˆˆπ’°U \in \mathcal{U}.
  6. Each member of a family is contained in the union of the family. So UβŠ†int⁑SU \subseteq \operatorname{int} S.

Idea

The interior is the union of all open sets that fit inside SS. Each of them lies in the union, and the union still fits inside SS. So the interior is the biggest open set that fits.

Reading path

To understand this page from the ground up, read these 30 pages in order. The foundations come first. The path takes the proof Proof that the interior is open.

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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…proofInterior is open, statement, no evidence levelInterior is openstatementProof that the interior is the largest open subset, 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
  29. Interior is open statement
  30. Proof that the interior is the largest open subset proof

Where this leads

6 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Interior_(topology)
About this page
TagF4T60QBCFC
URIm2://F4T60QBCFC
Kindproof
WritingW0 stub
MSC54A05
ProvesInterior is the largest open subset
UsesTopological space, Interior, Open set, Union of a family of sets, Interior is open
Changed4 November 2026

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