In
topology and
mathematics 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
∂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 and
manifold theory. However, frontier sometimes refers to a different set, which is the set of boundary points which are not actually in the set; that is,
S \
S.