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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 . 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, the term frontier has been used to describe the residue of S, namely S \ S (the set of boundary points not in S).Felix Hausdorff named the intersection of S with its boundary the border of S (the term boundary is used to refer to this set in Metric Spaces by E. T. Copson).
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:
These last two examples illustrate the fact that the boundary of a dense set with empty interior is its closure.
In the space of rational numbers with the usual topology (the subspace topology of ), the boundary of , where a is irrational, is empty.
The boundary of a set is a topological notion and may change if one changes the topology. For example, given the usual topology on , the boundary of a closed disk is the disk's surrounding circle: . If the disk is viewed as a set in with its own usual topology, i.e. , then the boundary of the disk is the disk itself: . If the disk is viewed as its own topological space (with the subspace topology of ), then the boundary of the disk is empty.
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, for any set S. The boundary operator thus satisfies a weakened kind of idempotence.
In discussing boundaries of manifolds or simplexes and their simplicial complexes, one often meets the assertion that the boundary of the boundary is always empty. Indeed, the construction of the singular homology 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.)