A subset
of a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
is called a regular open set if it is equal to the
interior
Interior may refer to:
Arts and media
* ''Interior'' (Degas) (also known as ''The Rape''), painting by Edgar Degas
* ''Interior'' (play), 1895 play by Belgian playwright Maurice Maeterlinck
* ''The Interior'' (novel), by Lisa See
* Interior de ...
of its
closure; expressed symbolically, if
or, equivalently, if
where
and
denote, respectively, the interior, closure and
boundary
Boundary or Boundaries may refer to:
* Border, in political geography
Entertainment
* ''Boundaries'' (2016 film), a 2016 Canadian film
* ''Boundaries'' (2018 film), a 2018 American-Canadian road trip film
*Boundary (cricket), the edge of the pla ...
of
[Steen & Seebach, p. 6]
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
Examples
If
has its usual
Euclidean topology
In mathematics, and especially general topology, the Euclidean topology is the natural topology induced on n-dimensional Euclidean space \R^n by the Euclidean metric.
Definition
The Euclidean norm on \R^n is the non-negative function \, \cdo ...
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.
Every regular open set is an
open set
In mathematics, open sets are a generalization of open intervals in the real line.
In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that a ...
and every regular closed set is a
closed set
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric spac ...
.
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
[Willard, "3D, Regularly open and regularly closed sets", p. 29] 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.
The collection of all regular open sets in
forms a
complete Boolean algebra
In mathematics, a complete Boolean algebra is a Boolean algebra in which every subset has a supremum (least upper bound). Complete Boolean algebras are used to construct Boolean-valued models of set theory in the theory of forcing. Every Boole ...
; the
join operation is given by
the
meet is
and the complement is
See also
*
*
*
*
Notes
References
* Lynn Arthur Steen and J. Arthur Seebach, Jr., ''Counterexamples in Topology''. Springer-Verlag, New York, 1978. Reprinted by Dover Publications, New York, 1995. (Dover edition).
*
General topology