Math 2.0

General topology Proposition

Closure is the smallest closed superset

Builds on ClosureClosure is closed

Statement

Let XX be a topological space and let S⊆XS \subseteq X. Then S⊆S‾S \subseteq \overline{S}, the set S‾\overline{S} is closed, and each closed set FF of XX with S⊆FS \subseteq F satisfies S‾⊆F\overline{S} \subseteq F.

Remarks

Here S‾\overline{S} is the closure of SS. That it is closed is the statement closure is closed. This is the dual of the statement that the interior is the largest open subset. If S⊆TS \subseteq T, then S‾⊆T‾\overline{S} \subseteq \overline{T}.

Reading path

To understand this page from the ground up, read these 44 pages in order. The foundations come first.

  1. Element definition
  2. Proposition concept
  3. Real numbers object
  4. Set definition
  5. Conjunction definition
  6. Disjunction definition
  7. Implication definition
  8. Negation definition
  9. Predicate definition
  10. Empty set object
  11. Family of sets definition
  12. Ordered pair definition
  13. Intersection definition
  14. Union definition
  15. Biconditional definition
  16. Truth table concept
  17. Existential quantifier definition
  18. Universal quantifier definition
  19. Cartesian product definition
  20. Propositional logic concept
  21. Subset definition
  22. Intersection of a family of sets definition
  23. Union of a family of sets definition
  24. Uniqueness quantifier definition
  25. Binary relation definition
  26. Universal set definition
  27. Topological space definition
  28. Tautology definition
  29. Complement definition
  30. Function definition
  31. Logical equivalence definition
  32. Metric space definition
  33. De Morgan's laws statement
  34. Open ball definition
  35. De Morgan's laws for sets statement
  36. Metric topology definition
  37. Open set definition
  38. Open sets of a metric space statement
  39. Closed set definition
  40. Closure definition
  41. Union of open sets is open statement
  42. Intersection of closed sets is closed statement
  43. Closure is closed statement
  44. Closure is the smallest closed superset statement

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Closure_(topology), Definitions
About this page
Tag0SHC6Z304Z
URIm2://0SHC6Z304Z
Kindstatement, proposition
WritingW0 stub
MSC54A05
StatusThis copy of the site has no build data. Run m2 status 0SHC6Z304Z.
UsesClosure, Closure is closed

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

Used by