in general, the boundary of a subset ''S'' of a topological space
''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
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
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.
A connected component
of the boundary of ''S'' is called a boundary component of ''S''.
There are several equivalent definitions for the boundary of a subset ''S'' of a topological space ''X'':
'' minus the interior
*the intersection of the closure of ''
'' with the closure of its complement
*the set of points
such that every neighborhood
contains at least one point of ''
'' and at least one point not of ''
Consider the real line
with the usual topology (i.e. the topology whose basis sets
are open interval
, the subset of rationals (with empty interior
). One has