Let A be a subset of a topological spaceMathworldPlanetmath X.

The union of all open sets contained in A is defined to be the interior of A. Equivalently, one could define the interior of A to the be the largest open set contained in A.

In this entry we denote the interior of A by int(A). Another common notation is A.

The exterior of A is defined as the union of all open sets whose intersectionMathworldPlanetmath with A is empty. That is, the exterior of A is the interior of the complement of A.

The interior of a set enjoys many special properties, some of which are listed below:

  1. 1.


  2. 2.

    int(A) is open

  3. 3.


  4. 4.


  5. 5.


  6. 6.

    A is open if and only if A=int(A)

  7. 7.


  8. 8.


  9. 9.

    AB implies that int(A)int(B)

  10. 10.

    int(A)=AA, where A is the boundary of A

  11. 11.



  • 1 S. Willard, General Topology, Addison-Wesley Publishing Company, 1970.
Title interior
Canonical name Interior
Date of creation 2013-03-22 12:48:20
Last modified on 2013-03-22 12:48:20
Owner yark (2760)
Last modified by yark (2760)
Numerical id 19
Author yark (2760)
Entry type Definition
Classification msc 54-00
Related topic Complement
Related topic ClosureMathworldPlanetmathPlanetmath
Related topic BoundaryInTopology
Defines exterior