Math 2.0

General topology Proposition

Interior is the largest open subset

Builds on InteriorInterior is open

Statement

Let XX be a topological space and let S⊆XS \subseteq X. Then int⁡S⊆S\operatorname{int} S \subseteq S, the set int⁡S\operatorname{int} S is open, and each open set UU of XX with U⊆SU \subseteq S satisfies U⊆int⁡SU \subseteq \operatorname{int} S.

Remarks

Here int⁡S\operatorname{int} S is the interior of SS. The openness is the statement interior is open. A consequence: SS is open if and only if int⁡S=S\operatorname{int} S = S. If S⊆TS \subseteq T, then int⁡S⊆int⁡T\operatorname{int} S \subseteq \operatorname{int} T.

Reading path

To understand this page from the ground up, read these 29 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 levelInteriordefinitionInterior is open, statement, no evidence levelInterior is openstatementInterior is the largest open subset, statement, no evidence levelInterior is the large…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.
  • 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. Interior is open statement
  29. Interior is the largest open subset statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Interior_(topology), Definitions
About this page
Tag0HAZEFH137
URIm2://0HAZEFH137
Kindstatement, proposition
WritingW0 stub
MSC54A05
StatusThis copy of the site has no build data. Run m2 status 0HAZEFH137.
UsesInterior, Interior is open

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