A subset of a
topological space is called a regular open set if it is equal to the
interior of its
closure; expressed symbolically, if or, equivalently, if where and denote, respectively, the interior, closure and
boundary of
[1]
A subset of is called a regular closed set if it is equal to the closure of its interior; expressed symbolically, if or, equivalently, if
[1]
Examples
If has its usual
Euclidean topology then the open set is not a regular open set, since Every
open interval in is a regular open set and every non-degenerate closed interval (that is, a closed interval containing at least two distinct points) is a regular closed set. A singleton is a closed subset of but not a regular closed set because its interior is the empty set so that
Properties
A subset of is a regular open set if and only if its complement in is a regular closed set.
[2] Every regular open set is an
open set and every regular closed set is a
closed set.
Each
clopen subset of (which includes and itself) is simultaneously a regular open subset and regular closed subset.
The interior of a closed subset of is a regular open subset of and likewise, the closure of an open subset of is a regular closed subset of
[2] The intersection (but not necessarily the union) of two regular open sets is a regular open set. Similarly, the union (but not necessarily the intersection) of two regular closed sets is a regular closed set.
[2]
The collection of all regular open sets in forms a
complete Boolean algebra; the
join operation is given by the
meet is and the complement is
See also
-
List of topologies – List of concrete topologies and topological spaces
-
Regular space – topological space in which a point and a closed set are, if disjoint, separable by neighborhoodsPages displaying wikidata descriptions as a fallback
-
Semiregular space
-
Separation axiom – Axioms in topology defining notions of "separation"
Notes
- ^
a
b Steen & Seebach, p. 6
- ^
a
b
c Willard, "3D, Regularly open and regularly closed sets", p. 29
References