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In
topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

and
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
in general, the boundary of a subset ''S'' of a
topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
''X'' is the set of points which can be approached both from ''S'' and from the outside of ''S''. More precisely, it is the set of points in the closure of ''S'' 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 ''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 bd(''S''), fr(''S''), 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
manifolds The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surface. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of suc ...

. 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 ''S'' is called a boundary component of ''S''.

# Common definitions

There are several equivalent definitions for the boundary of a subset ''S'' of a topological space ''X'': *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 $S$: $\partial S := \bar \setminus S^\circ$ *the intersection of the closure of ''$S$'' with the closure of its
complement A complement is often something that completes something else, or at least adds to it in some useful way. Thus it may be: * Complement (linguistics), a word or phrase having a particular syntactic role ** Subject complement, a word or phrase addi ...
: $\partial S := \overline \cap \overline$ *the set of points $p \in X$ such that every
neighborhood A neighbourhood (British English British English (BrE) is the standard dialect of the English language English is a West Germanic languages, West Germanic language first spoken in History of Anglo-Saxon England, early medieval ...
of $p$ contains at least one point of ''$S$'' and at least one point not of ''$S$'': $\partial S := \$

# Examples

Consider the real line $\R$ with the usual topology (i.e. the topology whose basis sets are
open interval In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
s) and $\Q$, the subset of rationals (with empty
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) ...
). One has * * $\partial \varnothing= \varnothing$ * $\partial \Q = \R$ * $\partial \left(\Q \cap$
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=
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/math> These last two examples illustrate the fact that the boundary of a
dense set In topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical stru ...
with empty interior is its closure. In the space of rational numbers with the usual topology (the
subspace topologyIn 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 relative ...

of $\R$), the boundary of $\left(-\infty, a\right)$, where ''a'' is irrational, is empty. The boundary of a set is a
topological s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structu ...

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 = \$ is the disk's surrounding circle: $\partial \Omega = \$. If the disk is viewed as a set in $\R^3$ with its own usual topology, i.e. $\Omega = \$, 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.

# Properties

* The boundary of a set is closed. * 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. * 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 the boundary of the complement of the set: $\partial S = \partial\left(S^c\right)$. * The interior of the boundary of a closed set is the empty set. * If $U$ is a
dense The density (more precisely, the volumetric mass density; also known as specific mass), of a substance is its mass Mass is both a property Property (''latin: Res Privata'') in the Abstract and concrete, abstract is what belongs to or ...
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 Australian ...
subset of $X$ then $\partial U = X \setminus U.$ Hence: * ("Trichotomy") Given a set $S$, a point lies in exactly one of the sets $\overset$, $\partial S$, and $\left(X\setminus S\right)^$. *''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 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 Australian ...
if and only if it is disjoint from its boundary. * The closure of a set equals the union of the set with its boundary: $\overline = S \cup \partial S$. * The boundary of a set is empty if and only if the set is both closed and open (that is, a
clopen set upright=1.3, A Graph (discrete mathematics), graph with several clopen sets. Each of the three large pieces (i.e. connected component (topology), components) is a clopen set, as is the union of any two or all three. In topology, a clopen set (a po ...
). * The interior of the boundary of the closure of a set is the empty set. :::: :''Conceptual
Venn diagram A Venn diagram is a widely used diagram A diagram is a symbolic representation Representation may refer to: Law and politics *Representation (politics) Political representation is the activity of making citizens "present" in public policy ...
showing the relationships among different points of a subset $S$ of Rn. A = set of
limit point In mathematics, a limit point (or cluster point or accumulation 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 def ...
s of $S,$ B = set of boundary points of $S,$ area shaded green = set of
interior point In mathematics, specifically in general topology, topology, the interior of a subset of a topological space is the Union (set theory), union of all subsets of that are Open set, open in . A point that is in the interior of is an interior point ...

s of $S,$ area shaded yellow = set of
isolated point 400px, "0" is an isolated point of A = ∪ , 2In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus ...
s of $S,$ areas shaded black = empty sets. Every point of $U$ is either an interior point or a boundary point. Also, every point of $U$ 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, 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 operations Operation or Operations may refer to: Science and technology * Surgical operation Surgery ''cheirourgikē'' (composed of χείρ, "hand", and ἔργον, "work"), via la, chirurgiae, m ...
. In discussing boundaries of
manifold The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surface. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of su ...

s or
simplex In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given space. For e ...

es and their
simplicial complex In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
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 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 ...
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.)