For a different notion of boundary related to manifolds, see that article.

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. Notations used for boundary of a set S include bd(S), fr(S), and $\backslash partial\; S$.

There are several common (and equivalent) definitions to the boundary of S:

• the intersection of the closure of S with the closure of its complement:

$\backslash partial\; S\; =\; \backslash bar\{S\}\; \backslash bigcap\; \backslash overline\{\; (X\; \backslash setminus\; S)\}.$

• the closure of S without the interior of S: $\backslash partial\; S\; =\; \backslash bar\{S\}\backslash setminus\; S^o$.
• a point p in X is a boundary point of S if every neighborhood of p contains at least one point of S and at least one point not in S. The boundary of S is the set of all boundary points of S.

Examples

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Consider the real line R with the usual topology (i.e. the topology whose basis sets are open intervals). One has

• $\backslash partial\; (0,5)\; =\; \backslash partial=\; \backslash partial\; (0,5=\; \backslash \{0,5\; \backslash \}\backslash ,$
• $\backslash partial\; \backslash emptyset\; =\; \backslash emptyset$
• $\backslash partial\; \backslash mathbb\{Q\}\; =\; \backslash mathbb\{R\}$
• $\backslash partial\; \backslash big(\backslash mathbb\{Q\}\backslash cap\backslash left=\; \backslash left[0,1\backslash right$

These last two examples illustrate the fact that the boundary of a dense set with empty interior is its closure.

One should keep in mind the boundary of a set is a topological notion, therefore, one changes the topology, the set boundary may change. For example, given the usual topology on R², and the closed disk

Ω={(x, y): x²+y² ≤ 1},

one has that ∂Ω = {(x, y) | x²+y² = 1} (the boundary of the disk is its surrounding circle).

In the same time, if that disk is viewed as a set in R³ with its own usual topology, that is,

Ω={(x, y, 0): x²+y² ≤ 1},

then ∂Ω = Ω (the boundary of the disk is the disk itself).

And lastly, if this disk is viewed as its own topological space (with the induced topology), then the boundary of the disk is empty.

Properties

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• The boundary of a set is closed.
• The boundary of a set is the boundary of the complement of the set: $\backslash partial\; S\; =\; \backslash partial\backslash bar\{S\}$.

Hence:

• p is a boundary point of a set if and only if every neighborhood of p contains at least one point in the set and at least one point not in the set.
• A set is closed if and only if it contains its boundary, and open if and only if it is disjoint from its boundary.
• The closure of a set equals the union of the set with its boundary. $\backslash bar\{S\}\; =\; S\; \backslash bigcup\backslash partial\; S$.
• The boundary of a set is empty if and only if the set is both closed and open (that is, a clopen set).
• In $\backslash mathbb\{R\}^n$, every closed set is the boundary of some set.

Conceptual Venn diagram showing the relationships among different points of set S. A = set of accumulation points of S, B = set of boundary points of S, area shaded green = set of interior points of S, area shaded yellow = set of isolated points of S, areas shaded black = empty sets. Every point of S is either an interior point or a boundary point. Also, every point of S is either an accumulation point or an isolated point. Likewise, every boundary point of S is either an accumulation point or an isolated point. Isolated points are always boundary points.

Boundary of a boundary

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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. See the discussion of boundary in topological manifold for more details.

General topology

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