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 ( | * 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> | :<math>(X - A)^{\circ} = X - \overline{A};~~ \overline{X-A} = X - A^{\circ}.</math> | ||
Latest revision as of 15:44, 7 February 2009
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.
