Interior (topology): Difference between revisions

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In [[mathematics]], the '''interior''' of a subset ''A'' of a [[topological space]] ''X'' is the [[union]] of all [[open set]]s in ''X'' that are [[subset]]s of ''A''.  It is usually denoted by <math>A^{\circ}</math>.  It may equivalently be defined as the set of all points in ''A'' for which ''A'' is a [[neighbourhood (topology)|neighbourhood]].
In [[mathematics]], the '''interior''' of a subset ''A'' of a [[topological space]] ''X'' is the [[union]] of all [[open set]]s in ''X'' that are [[subset]]s of ''A''.  It is usually denoted by <math>A^{\circ}</math>.  It may equivalently be defined as the set of all points in ''A'' for which ''A'' is a [[neighbourhood (topology)|neighbourhood]].


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* Interior is [[idempotence|idempotent]]: <math>A^{{\circ}{\circ}} = A^{\circ}</math>.
* Interior is [[idempotence|idempotent]]: <math>A^{{\circ}{\circ}} = A^{\circ}</math>.
* 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 (topology)|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>[[Category:Suggestion Bot Tag]]

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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.