
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
. 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 ''
'' 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
:
*the intersection of the closure of ''
'' 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 ...
:
*the set of points
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
contains at least one point of ''
'' and at least one point not of ''
'':
Examples

Consider the real line
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
, 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
*
*
*
*