In topology and mathematics in general, the boundary of a subset S of a topological space X is the set of points in the closure of S not belonging to the interior of S. An element of the boundary of S is called a boundary point of S. The term boundary operation refers to finding or taking the boundary of a set. Notations used for boundary of a set S include
Some authors (for example, Willard in General Topology) use the term frontier instead of boundary in an attempt to avoid confusion with a different definition used in algebraic topology and the theory of manifolds. Despite widespread acceptance of the meaning of the terms boundary and frontier, they have sometimes been used to refer to other sets. For example, Metric Spaces by E. T. Copson uses the term boundary to refer to Hausdorff's border, which is defined as the intersection of a set with its boundary. Hausdorff also introduced the term residue, which is defined as the intersection of a set with the closure of the border of its complement.
Definitions
There are several equivalent definitions for the boundary of a subset
S
⊆
X
{\displaystyle S\subseteq X}
of a topological space
X
,
{\displaystyle X,}
which will be denoted by
∂
X
S
,
{\displaystyle \partial _{X}S,}
or simply
∂
S
{\displaystyle \partial S}
if
X
{\displaystyle X}
is understood:
It is the closure of
S
{\displaystyle S}
minus the interior of
S
{\displaystyle S}
in
X
Examples
Consider the real line
R
{\displaystyle \mathbb {R} }
with the usual topology (that is, the topology whose basis sets are open intervals) and
Q
,
{\displaystyle \mathbb {Q} ,}
the subset of rational numbers (whose topological interior in
R
{\displaystyle \mathbb {R} }
is empty). Then in
R
{\displaystyle \mathbb {R} }
we have
∂
(
0
,
5
)
=
∂
[
0
,
5
)
=
∂
(
0
Properties
The boundary of a set is closed; this follows from the formula
{\displaystyle \partial S~:=~\{p\in X:{\text{ for every neighborhood }}O{\text{ of }}p,\ O\cap S\neq \varnothing \,{\text{ and }}\,O\cap (X\setminus S)\neq \varnothing \}.}
It is all points in
X
{\displaystyle X}
which are not in either the interior or exterior of
A boundary point of a set is any element of that set's boundary. The boundary
∂
X
S
{\displaystyle \partial _{X}S}
defined above is sometimes called the set's topological boundary to distinguish it from other similarly named notions such as the boundary of a manifold with boundary or the boundary of a manifold with corners, to name just a few examples.
A connected component of the boundary of S is called a boundary component of S.
These last two examples illustrate the fact that the boundary of a dense set with empty interior is its closure. They also show that it is possible for the boundary
∂
S
{\displaystyle \partial S}
of a subset
S
{\displaystyle S}
to contain a non-empty open subset of
X
:=
R
{\displaystyle X:=\mathbb {R} }
; that is, for the interior of
∂
S
{\displaystyle \partial S}
in
X
{\displaystyle X}
to be non-empty. However, a closed subset's boundary always has an empty interior.
The notation
∂
X
S
{\displaystyle \partial _{X}S}
is used because the boundary of a set
S
{\displaystyle S}
crucially depends on the surrounding topological space
The interior of the boundary of a closed set is empty.
Consequently, the interior of the boundary of the closure of a set is empty.
The interior of the boundary of an open set is also empty.
Consequently, the interior of the boundary of the interior of a set is empty.
In particular, if
S
⊆
X
{\displaystyle S\subseteq X}
is a closed or open subset of
X
{\displaystyle X}
then there does not exist any nonempty subset
U
⊆
∂
X
S
{\displaystyle U\subseteq \partial _{X}S}
such that
U
{\displaystyle U}
is open in
X
.
{\displaystyle X.}
This fact is important for the definition and use of nowhere dense subsets, meager subsets, and Baire spaces.
A set is the boundary of some open set if and only if it is closed and nowhere dense.
The boundary of a set is empty if and only if the set is both closed and open (that is, a clopen set).
for any set
S
.
{\displaystyle S.}
The boundary operator thus satisfies a weakened kind of idempotence.
In discussing boundaries of manifolds or simplexes and their simplicial complexes, one often meets the assertion that the boundary of the boundary is always empty. Indeed, the construction of the singular homology rests critically on this fact. The explanation for the apparent incongruity is that the topological boundary (the subject of this article) is a slightly different concept from the boundary of a manifold or of a simplicial complex. For example, the boundary of an open disk viewed as a manifold is empty, as is its topological boundary viewed as a subset of itself, while its topological boundary viewed as a subset of the real plane is the circle surrounding the disk. Conversely, the boundary of a closed disk viewed as a manifold is the bounding circle, as is its topological boundary viewed as a subset of the real plane, while its topological boundary viewed as a subset of itself is empty. In particular, the topological boundary depends on the ambient space, while the boundary of a manifold is invariant.
In topology and mathematics in general, the boundary of a subset S of a topological space X is the set of points in the closure of S not belonging to the interior of S. An element of the boundary of S is called a boundary point of S. The term boundary operation refers to finding or taking the boundary of a set. Notations used for boundary of a set S include and .
