Boundary (topology)

This article is about boundaries in general topology. For the boundary of a manifold, see boundary of a manifold.
A set (in light blue) and its boundary (in dark blue).

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.

A connected component of the boundary of S is called a boundary component of S.

If the set consists of discrete points only, then the set has only a boundary and no interior.

Common definitions

There are several common (and equivalent) definitions to the boundary of a subset S of a topological space X:


Boundary of hyperbolic components of Mandelbrot set

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 , 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 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 subspace topology of R2), then the boundary of the disk is empty.



Conceptual Venn diagram showing the relationships among different points of a subset S of Rn. A = set of limit 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

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, ∂∂S = ∂∂∂S 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, while its boundary in the sense of topological space is the circle surrounding the disk.

See also


  1. Mendelson, Bert (1990) [1975]. Introduction to Topology (Third ed.). Dover. p. 86. ISBN 0-486-66352-3. Corollary 4.15 For each subset A, Brdy (A) is closed.

Further reading

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