Interior (topology): Difference between revisions

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imported>Richard Pinch
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imported>Richard Pinch
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* Interior [[distributivity|distributes]] over finite [[intersection]]: <math>(A \cap B)^{\circ} = A^{\circ} \cap B^{\circ}</math>.  
* Interior [[distributivity|distributes]] over finite [[intersection]]: <math>(A \cap B)^{\circ} = A^{\circ} \cap B^{\circ}</math>.  
* The complement of the [[closure (mathematics)|closure]] of a set in ''X'' is the interior of the complement of that set; the complement of the interior of a set in ''X'' is the closure of the complement of that set.
* The complement of the [[closure (mathematics)|closure]] of a set in ''X'' is the interior of the complement of that set; the complement of the interior of a set in ''X'' is the closure of the complement of that set.
:<math>(X - A)^{\circ} = X - \overline{A};~~ \overline{X-A} = X - A^{\circ}.</math>

Revision as of 16:42, 27 December 2008

In mathematics, the interior of a subset A of a topological space X is the union of all open sets in X that are subsets of A. It is usually denoted by . It may equivalently be defined as the set of all points in A for which A is a neighbourhood.

Properties

  • A set contains its interior, .
  • The interior of a open set G is just G itself, .
  • Interior is idempotent: .
  • Interior distributes over finite intersection: .
  • The complement of the closure of a set in X is the interior of the complement of that set; the complement of the interior of a set in X is the closure of the complement of that set.