boundary (topology)

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OR:

In
topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
and
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
in general, the boundary of a subset of a
topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
is the set of points in the closure of not belonging to the
interior Interior may refer to: Arts and media * Interior (Degas), ''Interior'' (Degas) (also known as ''The Rape''), painting by Edgar Degas * Interior (play), ''Interior'' (play), 1895 play by Belgian playwright Maurice Maeterlinck * The Interior (novel ...
of . An element of the boundary of is called a boundary point of . The term boundary operation refers to finding or taking the boundary of a set. Notations used for boundary of a set include $\operatorname\left(S\right), \operatorname\left(S\right),$ and $\partial S$. 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 Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...
and the theory of
manifold In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
s. 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. A connected component of the boundary of is called a boundary component of .

# Common definitions

There are several equivalent definitions for the of a subset $S \subseteq X$ of a topological space $X,$ which will be denoted by $\partial_X S,$ $\operatorname_X S,$ or simply $\partial S$ if $X$ is understood:
1. It is the closure of $S$ minus the
interior Interior may refer to: Arts and media * Interior (Degas), ''Interior'' (Degas) (also known as ''The Rape''), painting by Edgar Degas * Interior (play), ''Interior'' (play), 1895 play by Belgian playwright Maurice Maeterlinck * The Interior (novel ...
of $S$ in $X$: $\partial S ~:=~ \overline \setminus \operatorname_X S$ where $\overline = \operatorname_X S$ denotes the closure of $S$ in $X$ and $\operatorname_X S$ denotes the topological interior of $S$ in $X.$
2. It is the intersection of the closure of $S$ with the closure of its
complement A complement is something that completes something else. Complement may refer specifically to: The arts * Complement (music), an interval that, when added to another, spans an octave ** Complement (music)#Aggregate complementation, Aggregate c ...
: $\partial S ~:=~ \overline \cap \overline$
3. It is the set of points $p \in X$ such that every
neighborhood A neighbourhood (British English British English (BrE, en-GB, or BE) is, according to Oxford Dictionaries, " English as used in Great Britain, as distinct from that used elsewhere". More narrowly, it can refer specifically to the En ...
of $p$ contains at least one point of $S$ and at least one point not of $S$: $\partial S ~:=~ \.$
A of a set refers to any element of that set's boundary. The boundary $\partial_X S$ defined above is sometimes called the set's to distinguish it from other similarly named notions such as the boundary of a
manifold with boundary In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a Ne ...
or the boundary of a manifold with corners, to name just a few examples.

# Properties

The closure of a set $S$ equals the union of the set with its boundary: $\overline = S \cup \partial_X S$ where $\overline = \operatorname_X S$ denotes the closure of $S$ in $X.$ A set is closed if and only if it contains its boundary, and open if and only if it is disjoint from its boundary. The boundary of a set is closed; this follows from the formula $\partial_X S ~:=~ \overline \cap \overline,$ which expresses $\partial_X S$ as the intersection of two closed subsets of $X.$ ("Trichotomy") Given any subset $S \subseteq X,$ each point of $X$ lies in exactly one of the three sets $\operatorname_X S, \partial_X S,$ and $\operatorname_X \left(X \setminus S\right).$ Said differently, $X ~=~ \left(\operatorname_X S\right) \;\cup\; \left(\partial_X S\right) \;\cup\; \left(\operatorname_X (X \setminus S)\right)$ and these three sets are pairwise disjoint. Consequently, if these set are not emptyThe condition that these sets be non-empty is needed because sets in a partition are by definition required to be non-empty. then they form a partition of $X.$ A point $p \in X$ 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. The boundary of the interior of a set as well as the boundary of the closure of a set are both contained in the boundary of the set.

''Conceptual
Venn diagram A Venn diagram is a widely used diagram A diagram is a symbolic Depiction, representation of information using Visualization (graphics), visualization techniques. Diagrams have been used since prehistoric times on Cave painting, walls of ca ...
showing the relationships among different points of a subset $S$ of $\R^n.$ $A$ = set of
limit point In mathematics, a limit point, accumulation point, or cluster point of a set S in a topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined ...
s of $S,$ $B =$ set of boundary points of $S,$ area shaded green = set of interior points of $S,$ area shaded yellow = set of isolated points 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.''

# Examples

## Characterizations and general examples

The boundary of a set is equal to the boundary of the set's complement: $\partial_X S = \partial_X (X \setminus S).$ A set $U$ is a dense
open Open or OPEN may refer to: Music * Open (band) Open is a band. Background Drummer Pete Neville has been involved in the Sydney/Australian music scene for a number of years. He has recently completed a Masters in screen music at the Australia ...
subset of $X$ if and only if $\partial_X U = X \setminus U.$ The interior of the boundary of a closed set is the empty set.Let $S$ be a closed subset of $X$ so that $\overline = S$ and thus also $\partial_X S := \overline \setminus \operatorname_X S = S \setminus \operatorname_X S.$ If $U$ is an open subset of $X$ such that $U \subseteq \partial_X S$ then $U \subseteq S$ (because $\partial_X S \subseteq S$) so that $U \subseteq \operatorname_X S$ (because by definition, $\operatorname_X S$ is the largest open subset of $X$ contained in $S$). But $U \subseteq \partial_X S = S \setminus \operatorname_X S$ implies that $U \cap \operatorname_X S = \varnothing.$ Thus $U$ is simultaneously a subset of $\operatorname_X S$ and disjoint from $\operatorname_X S,$ which is only possible if $U = \varnothing.$ Q.E.D. Consequently, the interior of the boundary of the closure of a set is the empty set. The interior of the boundary of an open set is also the empty set.Let $S$ be an open subset of $X$ so that $\partial_X S := \overline \setminus \operatorname_X S = \overline \setminus S.$ Let $U := \operatorname_X \left\left(\partial_X S\right\right)$ so that $U = \operatorname_X \left\left(\partial_X S\right\right) \subseteq \partial_X S = \overline \setminus S,$ which implies that $U \cap S = \varnothing.$ If $U \neq \varnothing$ then pick $u \in U,$ so that $u \in U \subseteq \partial_X S \subseteq \overline.$ Because $U$ is an open neighborhood of $u$ in $X$ and $u \in \overline,$ the definition of the
topological closure In topology, the closure of a subset of points in a topological space consists of all Topology glossary#P, points in together with all limit points of . The closure of may equivalently be defined as the Union (set theory), union of and its Bo ...
$\overline$ implies that $U \cap S \neq \varnothing,$ which is a contradiction. $\blacksquare$ Alternatively, if $S$ is open in $X$ then $X \setminus S$ is closed in $X,$ so that by using the general formula $\partial_X S = \partial_X \left(X \setminus S\right)$ and the fact that the interior of the boundary of a closed set (such as $X \setminus S$) is empty, it follows that $\operatorname_X \partial_X S = \operatorname_X \partial_X \left(X \setminus S\right) = \varnothing.$ $\blacksquare$
Consequently, the interior of the boundary of the interior of a set is the empty set. In particular, if $S \subseteq X$ is a closed or open subset of $X$ then there does not exist any non-empty subset $U \subseteq \partial_X S$ such that $U$ is also an open subset of $X.$ This fact is important for the definition and use of nowhere dense subsets, meager subsets, and
Baire space In mathematics, a topological space X is said to be a Baire space if countable unions of closed sets with empty interior (topology), interior also have empty interior. According to the Baire category theorem, compact Hausdorff spaces and complete ...
s. 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 In topology, a clopen set (a portmanteau of closed-open set) in a topological space is a set which is both open set, open and closed set, closed. That this is possible may seem counter-intuitive, as the common meanings of and are antonyms, but ...
).

## Concrete examples

Consider the real line $\R$ with the usual topology (that is, the topology whose basis sets are
open interval In mathematics, a (real) interval is a set (mathematics), set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers satisfying is an interval which contains , , and all numbers ...
s) and $\Q,$ the subset of rational numbers (whose topological interior in $\R$ is empty). Then * * $\partial \varnothing= \varnothing$ * $\partial \Q = \R$ *
dense set In topology and related areas of mathematics, a subset ''A'' of a topological space ''X'' is said to be dense in ''X'' if every point of ''X'' either belongs to ''A'' or else is arbitrarily "close" to a member of ''A'' — for instance, the ra ...
with empty interior is its closure. They also show that it is possible for the boundary $\partial S$ of a subset $S$ to contain a non-empty open subset of $X := \R$; that is, for the interior of $\partial S$ in $X$ to be non-empty. However, a subset's boundary always has an empty interior. In the space of rational numbers with the usual topology (the
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 Topological_space#Definitions, topology induced from that of ''X'' called the subspace topology (or the relat ...
of $\R$), the boundary of $\left(-\infty, a\right),$ where $a$ is irrational, is empty. The boundary of a set is a
topological In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
notion and may change if one changes the topology. For example, given the usual topology on $\R^2,$ the boundary of a closed disk $\Omega = \left\$ is the disk's surrounding circle: $\partial \Omega = \left\.$ If the disk is viewed as a set in $\R^3$ with its own usual topology, that is, $\Omega = \left\,$ then the boundary of the disk is the disk itself: $\partial \Omega = \Omega.$ If the disk is viewed as its own topological space (with the subspace topology of $\R^2$), then the boundary of the disk is empty.

## Boundary of an open ball vs. its surrounding sphere

This example demonstrates that the topological boundary of an open ball of radius $r > 0$ is necessarily equal to the corresponding sphere of radius $r$ (centered at the same point); it also shows that the closure of an open ball of radius $r > 0$ is necessarily equal to the closed ball of radius $r$ (again centered at the same point). Denote the usual
Euclidean metric In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two Point (geometry), points. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theo ...
on $\R^2$ by $d((a, b), (x, y)) := \sqrt$ which induces on $\R^2$ the 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 distance, Euclidean metric. Definition The Euclidean norm on \R^n is the non-negative f ...
. Let $X \subseteq \R^2$ denote the union of the $y$-axis $Y := \ \times \R$ with the unit circle $S^1 := \left\ = \left\$ centered at the origin $\mathbf := \left(0, 0\right) \in \R^2$; that is, $X := Y \cup S^1,$ which is a
topological subspace 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 Topological_space#Definitions, topology induced from that of ''X'' called the subspace topology (or the relat ...
of $\R^2$ whose topology is equal to that induced by the (restriction of) the metric $d.$ In particular, the sets $Y, S^1, Y \cap S^1 = \,$ and
metric space In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functio ...
$\left(X, d\right)$ will be considered (and not its superspace $\left(\R^2, d\right)$); this being a
path-connected In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union (set theory), union of two or more disjoint set, disjoint Empty set, non-empty open (topology), open subsets. Conn ...
and locally path-connected
complete metric space In mathematical analysis, a metric space is called complete (or a Cauchy space) if every Cauchy sequence#In a metric space, Cauchy sequence of points in has a Limit of a sequence, limit that is also in . Intuitively, a space is complete if ther ...
. Denote the open ball of radius $r > 0$ in $\left(X, d\right)$ by $B_r := \left\$ so that when $r = 1$ then $B_1 = \ \times (-1, 1)$ is the open sub-interval of the $y$-axis strictly between $y = -1$ and $y = 1.$ The unit sphere in $\left(X, d\right)$ ("unit" meaning that its radius is $r = 1$) is $\left\ = S^1$ while the closed unit ball in $\left(X, d\right)$ is the union of the open unit ball and the unit sphere centered at this same point: However, the topological boundary $\partial_X B_1$ and topological closure $\operatorname_X B_1$ in $X$ of the open unit ball $B_1$ are: In particular, the open unit ball's topological boundary $\partial_X B_1 = \$ is a subset of the unit sphere $\left\ = S^1$ in $\left(X, d\right).$ And the open unit ball's topological closure $\operatorname_X B_1 = B_1 \cup \$ is a proper subset of the closed unit ball in $\left(X, d\right).$ The point $\left(1, 0\right) \in X,$ for instance, cannot belong to $\operatorname_X B_1$ because there does not exist a sequence in $B_1 = \ \times \left(-1, 1\right)$ that converges to it; the same reasoning generalizes to also explain why no point in $X$ outside of the closed sub-interval

# Boundary of a boundary

For any set $S, \partial S \supseteq \partial\partial S,$ where $\,\supseteq\,$ denotes the
superset In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
with equality holding if and only if the boundary of $S$ has no interior points, which will be the case for example if $S$ is either closed or open. Since the boundary of a set is closed, $\partial \partial S = \partial \partial \partial S$ for any set $S.$ The boundary operator thus satisfies a weakened kind of
idempotence Idempotence (, ) is the property of certain operation (mathematics), operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence ...
. In discussing boundaries of
manifold In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
s or
simplex In geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of ...
es and their
simplicial complex In mathematics, a simplicial complex is a Set (mathematics), set composed of Point (geometry), points, line segments, triangles, and their Simplex, ''n''-dimensional counterparts (see illustration). Simplicial complexes should not be confused with ...
es, one often meets the assertion that the boundary of the boundary is always empty. Indeed, the construction of the
singular homology In algebraic topology, singular homology refers to the study of a certain set of algebraic invariants of a topological space ''X'', the so-called homology groups H_n(X). Intuitively, singular homology counts, for each dimension ''n'', the ''n''-d ...
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.