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Boundary (topology)
 
 
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For a different notion of boundary related to manifold
Manifold

In mathematics, more specifically topology, a manifold is a topological space in which every point has a neighborhood which "resembles" Euclidean space....
s, see that article.
In topology
Topology

Topology is a major area of mathematics that has emerged through the development of concepts from geometry and set theory, such as those of space, dimension, shape, transformation and others....
, the boundary of a subset S of a topological space
Topological space

Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connected space, and Continuous function ....
 X is the set of points which can be approached both from S and from the outside of S. More formally, it is the set of points in the closure
Closure (topology)

In mathematics, the closure of a set S consists of all Topology glossary#Ps which are intuitively "close to S". A point which is in the closure of S is a adherent point of S....
 of S, not belonging to the interior
Interior (topology)

In mathematics, the interior of a set S consists of all Topology glossary#Ps of S that are intuitively "not on the edge of S". A point that is in the interior of S is an interior point of S....
 of S. An element of the boundary of S is called a boundary point of S.






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For a different notion of boundary related to manifold
Manifold

In mathematics, more specifically topology, a manifold is a topological space in which every point has a neighborhood which "resembles" Euclidean space....
s, see that article.
In topology
Topology

Topology is a major area of mathematics that has emerged through the development of concepts from geometry and set theory, such as those of space, dimension, shape, transformation and others....
, the boundary of a subset S of a topological space
Topological space

Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connected space, and Continuous function ....
 X is the set of points which can be approached both from S and from the outside of S. More formally, it is the set of points in the closure
Closure (topology)

In mathematics, the closure of a set S consists of all Topology glossary#Ps which are intuitively "close to S". A point which is in the closure of S is a adherent point of S....
 of S, not belonging to the interior
Interior (topology)

In mathematics, the interior of a set S consists of all Topology glossary#Ps of S that are intuitively "not on the edge of S". A point that is in the interior of S is an interior point of S....
 of S. An element of the boundary of S is called a boundary point of S. S is boundaryless when it contains no boundary, which is to say no boundary point (as distinct from the metric notion of unbounded set
Bounded set

In mathematical analysis and related areas of mathematics, a Set is called bounded, if it is, in a certain sense, of finite size. Conversely a set which is not bounded is called unbounded....
). Notations used for boundary of a set S include bd(S), fr(S), and ?S. Some authors (for example Willard, in General Topology) use the term 'frontier', instead of boundary in an attempt to avoid confusion with the concept of boundary used in algebraic topology.

A of the boundary of S is called a boundary component of S.

Common definitions


There are several common (and equivalent) definitions to the boundary of a subset S of a topological space X:
  • the closure of S without the interior of S: ?S = S \ So.
  • the intersection of the closure of S with the closure of its complement
    Complement (set theory)

    In discrete mathematics and predominantly in set theory, a complement is a concept used in comparisons of sets to refer to the unique values of one set in relation to another....
    : ?S = S n (X \ S).
  • the set of points p of X such that every neighborhood of p contains at least one point of S and at least one point not of S.


Examples

Consider the real line R with the usual topology (i.e. the topology whose basis sets are open intervals). One has

  • ?(0,5) = ?[0,5) = ?(0,5] = ?[0,5] = 
  • ?Ø = Ø
  • ?Q = R
  • ?(Q n [0,1]) = [0,1]


These last two examples illustrate the fact that the boundary of a dense set
Dense set

In topology and related areas of mathematics, a subset A of a topological space X is called dense if, intuitively, any point in X can be "well-approximated" by points in A....
 with empty interior is its closure.

In the space of rational numbers with the usual topology (the subspace topology
Subspace topology

In topology and related areas of mathematics, a subspace of a topological space X is a subset S of X which is equipped with a natural topology induced from that of X called the subspace topology ....
 of R), the boundary of , where a is irrational, is empty.

The boundary of a set is a topological
Topology

Topology is a major area of mathematics that has emerged through the development of concepts from geometry and set theory, such as those of space, dimension, shape, transformation and others....
 notion and may change if one changes the topology. For example, given the usual topology on R2, the boundary of a closed disk O = is the disk's surrounding circle: ?O = . If the disk is viewed as a set in R3 with its own usual topology, i.e. O = , then the boundary of the disk is the disk itself: ?O = O. If the disk is viewed as its own topological space (with the induced topology), then the boundary of the disk is empty.

Properties

  • The boundary of a set is closed
    Closed set

    In topology and related branches of mathematics, a closed set is a Set whose complement is open set....
    .
  • The boundary of a set is the boundary of the complement of the set: ?S = ?(SC).
Hence:
  • p is a boundary point of a set if and only if every neighborhood of p contains at least one point in the set and at least one point not in the set.
  • A set is closed if and only if it contains its boundary, and open
    Open set

    In metric topology and related fields of mathematics, a Set U is called open if, intuitively speaking, starting from any point x in U one can move by a small amount in any direction and still be in the set U....
     if and only if it is disjoint from its boundary.
  • The closure of a set equals the union of the set with its boundary. S = S ? ?S.
  • The boundary of a set is empty if and only if the set is both closed and open (that is, a clopen set
    Clopen set

    In topology, a clopen set in a topological space is a set which is both open set and closed set....
    ).
  • In Rn, every closed set is the boundary of some set.


Accumulationandboundarypointsofs
:Concept
Concept

A concept is a cognition unit of meaning— an abstraction idea or a mental symbol sometimes defined as a "unit of knowledge," built from other units which act as a concept's characteristics....
ual Venn diagram
Venn diagram

Venn diagrams or set diagrams are diagrams that show all hypothetically possible logical relations between a finite collection of Set . Venn diagrams were invented around 1880 by John Venn....
 showing the relationships among different points of a subset S of Rn. A = set of accumulation points of S, B = set of boundary points of S, area shaded green = set of interior points of S, area shaded yellow = set of isolated point
Isolated point

In topology, a branch of mathematics, a point x of a Set S is called an isolated point,if there exists a Neighborhood of x not containing other points of S....
s of S, areas shaded black = empty sets. Every point of S is either an interior point or a boundary point. Also, every point of S is either an accumulation point or an isolated point. Likewise, every boundary point of S is either an accumulation point or an isolated point. Isolated points are always boundary points.

Boundary of a boundary

For any set S, ?S ? ??S, with equality holding if and only if the boundary of S has no interior points. This is always true if S is either closed or open. Since the boundary of any set is closed, ??S = ???S for any set S. The boundary operator thus satisfies a weakened kind of idempotence
Idempotence

Idempotence describes the property of operations in mathematics and computer science which means that multiple applications of the operation does not change the result....
. In particular, the boundary of the boundary of a set will usually be nonempty.

In discussing boundaries of manifold
Manifold

In mathematics, more specifically topology, a manifold is a topological space in which every point has a neighborhood which "resembles" Euclidean space....
s or simplex
Simplex

In geometry, a simplex or n-simplex is an n-dimensional analogue of a triangle. Specifically, a simplex is the convex hull of a set of affine transformation Point s in some Euclidean space of dimension n or higher ....
es and their simplicial complex
Simplicial complex

In mathematics, a simplicial complex is a topological space of a particular kind, constructed by "gluing together" Point s, line segments, triangles, and their n-dimensional counterparts ....
es, one often meets the assertion that the boundary of the boundary is always empty. Indeed, the construction of the singular homology
Singular homology

In algebraic topology, a branch of mathematics, singular homology refers to the study of a certain set of topological invariants of a topological space X, the so-called homology groups ....
 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 than the boundary of a manifold or of a simplicial complex. For example, the topological boundary of a closed disk viewed as a topological space is empty, while its boundary in the sense of manifolds is the circle surrounding the disk.

See also

  • See the discussion of boundary in topological manifold
    Topological manifold

    In mathematics, a topological manifold is a Hausdorff space topological space which looks locally like Euclidean space in a sense defined below....
     for more details.