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 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 \, \cdot\, ...
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 are su ...
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
[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 Boolea ...
; the
join Join may refer to:
* Join (law), to include additional counts or additional defendants on an indictment
*In mathematics:
** Join (mathematics), a least upper bound of sets orders in lattice theory
** Join (topology), an operation combining two top ...
operation is given by
the
meet
Meet may refer to:
People with the name
* Janek Meet (born 1974), Estonian footballer
* Meet Mukhi (born 2005), Indian child actor
Arts, entertainment, and media
* ''Meet'' (TV series), an early Australian television series which aired on ABC du ...
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