In topology, 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 formally, 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. S is boundaryless when it contains no boundary, which is to say no boundary point (as distinct from the metric notion of unbounded 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.
A connected component of the boundary of S is called a boundary component of S.
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There are several common (and equivalent) definitions to the boundary of a subset S of a topological space X:
Consider the real line R with the usual topology (i.e. the topology whose basis sets are open intervals). One has
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 R), the boundary of the set of numbers of which the square is less than 2 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 R2, the boundary of a closed disk Ω = {(x,y) | x2 + y2 ≤ 1} is the disk's surrounding circle: ∂Ω = {(x,y) | x2 + y2 = 1}. If the disk is viewed as a set in R3 with its own usual topology, i.e. Ω = {(x,y,0) | x2 + y2 ≤ 1}, then the boundary of the disk is the disk itself: ∂Ω = Ω. If the disk is viewed as its own topological space (with the induced topology), then the boundary of the disk is empty.
Hence:
For any set S, ∂S ⊇ ∂∂S, with equality holding if and only if the boundary of S has no interior points. This is always true if S is either closed or open. Since the boundary of any set is closed, ∂∂S = ∂∂∂S for any set S. The boundary operator thus satisfies a weakened kind of idempotence. In particular, the boundary of the boundary of a set will usually be nonempty.
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 than the boundary of a manifold or of a simplicial complex. For example, the topological boundary of a closed disk viewed as a topological space is empty, while its boundary in the sense of manifolds is the circle surrounding the disk.