Boundary (topology)

For a different notion of boundary related to manifolds, see that article.
A set (in light blue) and its boundary (in dark blue).

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 componenthttp://localhost../../../../articles/c/o/n/Connected_space.html#Formal_definition of the boundary of S is called a boundary component of S.

Contents

Common definitions

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

Examples

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 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.

Properties

Hence:

AccumulationAndBoundaryPointsOfS.PNG
Conceptual Venn diagram showing the relationships among different points of a subset S of Rn. 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

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 also

References