- Source: Regular open set
A subset
S
{\displaystyle S}
of a topological space
X
{\displaystyle X}
is called a regular open set if it is equal to the interior of its closure; expressed symbolically, if
Int
(
S
¯
)
=
S
{\displaystyle \operatorname {Int} ({\overline {S}})=S}
or, equivalently, if
∂
(
S
¯
)
=
∂
S
,
{\displaystyle \partial ({\overline {S}})=\partial S,}
where
Int
S
,
{\displaystyle \operatorname {Int} S,}
S
¯
{\displaystyle {\overline {S}}}
and
∂
S
{\displaystyle \partial S}
denote, respectively, the interior, closure and boundary of
S
.
{\displaystyle S.}
A subset
S
{\displaystyle S}
of
X
{\displaystyle X}
is called a regular closed set if it is equal to the closure of its interior; expressed symbolically, if
Int
S
¯
=
S
{\displaystyle {\overline {\operatorname {Int} S}}=S}
or, equivalently, if
∂
(
Int
S
)
=
∂
S
.
{\displaystyle \partial (\operatorname {Int} S)=\partial S.}
Examples
If
R
{\displaystyle \mathbb {R} }
has its usual Euclidean topology then the open set
S
=
(
0
,
1
)
∪
(
1
,
2
)
{\displaystyle S=(0,1)\cup (1,2)}
is not a regular open set, since
Int
(
S
¯
)
=
(
0
,
2
)
≠
S
.
{\displaystyle \operatorname {Int} ({\overline {S}})=(0,2)\neq S.}
Every open interval in
R
{\displaystyle \mathbb {R} }
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
{
x
}
{\displaystyle \{x\}}
is a closed subset of
R
{\displaystyle \mathbb {R} }
but not a regular closed set because its interior is the empty set
∅
,
{\displaystyle \varnothing ,}
so that
Int
{
x
}
¯
=
∅
¯
=
∅
≠
{
x
}
.
{\displaystyle {\overline {\operatorname {Int} \{x\}}}={\overline {\varnothing }}=\varnothing \neq \{x\}.}
Properties
A subset of
X
{\displaystyle X}
is a regular open set if and only if its complement in
X
{\displaystyle X}
is a regular closed set. Every regular open set is an open set and every regular closed set is a closed set.
Each clopen subset of
X
{\displaystyle X}
(which includes
∅
{\displaystyle \varnothing }
and
X
{\displaystyle X}
itself) is simultaneously a regular open subset and regular closed subset.
The interior of a closed subset of
X
{\displaystyle X}
is a regular open subset of
X
{\displaystyle X}
and likewise, the closure of an open subset of
X
{\displaystyle X}
is a regular closed subset of
X
.
{\displaystyle X.}
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
X
{\displaystyle X}
forms a complete Boolean algebra; the join operation is given by
U
∨
V
=
Int
(
U
∪
V
¯
)
,
{\displaystyle U\vee V=\operatorname {Int} ({\overline {U\cup V}}),}
the meet is
U
∧
V
=
U
∩
V
{\displaystyle U\land V=U\cap V}
and the complement is
¬
U
=
Int
(
X
∖
U
)
.
{\displaystyle \neg U=\operatorname {Int} (X\setminus U).}
See also
List of topologies – List of concrete topologies and topological spaces
Regular space – Property of topological space
Semiregular space
Separation axiom – Axioms in topology defining notions of "separation"
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. ISBN 0-486-68735-X (Dover edition).
Willard, Stephen (2004) [1970]. General Topology. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-43479-7. OCLC 115240.
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- Regular open set
- Open set
- Regular space
- Glossary of general topology
- Semiregular space
- Open and closed maps
- Regular
- Regular expression
- Regular measure
- Separation axiom
