regular open set

Let X be a topological spaceMathworldPlanetmath. A subset A of X is called a regular open set if A is equal to the interior of the closureMathworldPlanetmathPlanetmath of itself:


Clearly, every regular open set is open, and every clopen set is regular open.

Examples. Let be the real line with the usual topology (generated by open intervalsPlanetmathPlanetmath).

  • (a,b) is regular open whenever -<ab<.

  • (a,b)(b,c) is not regular open for -<abc< and ac. The interior of the closure of (a,b)(b,c) is (a,c).

If we examine the structureMathworldPlanetmath of int(A¯) a little more closely, we see that if we define




So an alternative definition of a regular open set is an open set A such that A=A.


  • For any AX, A is always open.

  • =X and X=.

  • AA= and AA is dense in X.

  • AB(AB) and AB=(AB).

  • It can be shown that if A is open, then A is regular open. As a result, following from the first property, int(A¯), being A, is regular open for any subset A of X.

  • In addition, if both A and B are regular open, then AB is regular open.

  • It is not true, however, that the union of two regular open sets is regular open, as illustrated by the second example above.

  • It can also be shown that the set of all regular open sets of a topological space X forms a Boolean algebraMathworldPlanetmath under the following set of operationsMathworldPlanetmath:

    1. (a)

      1=X and 0=,

    2. (b)


    3. (c)

      ab=(ab), and

    4. (d)


    This is an example of a Boolean algebra coming from a collectionMathworldPlanetmath of subsets of a set that is not formed by the standard set operations union , intersectionMathworldPlanetmath , and complementation .

The definition of a regular open set can be dualized. A closed setPlanetmathPlanetmath A in a topological space is called a regular closed set if A=int(A)¯.


  • 1 P. Halmos (1970). Lectures on Boolean Algebras, Springer.
  • 2 S. Willard (1970). General Topology, Addison-Wesley Publishing Company.
Title regular open set
Canonical name RegularOpenSet
Date of creation 2013-03-22 15:04:03
Last modified on 2013-03-22 15:04:03
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 9
Author CWoo (3771)
Entry type Definition
Classification msc 06E99
Synonym regularly open
Synonym regularly closed
Synonym regularly closed set
Defines regular open
Defines regular closed