Boundary (topology)

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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. 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.

A connected component 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:

  • the closure of S without the interior of S: ∂S = S \ So.
  • the intersection of the closure of S with the closure of its complement: ∂S = S ∩ (X \ S).
  • the set of points p of X such that every neighborhood of p contains at least one point of S and at least one point not of S.

Examples

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

  • ∂(0,5) = ∂[0,5) = ∂(0,5] = ∂[0,5] = {0,5}
  • ∂∅ = ∅
  • Q = R
  • ∂(Q ∩ [0,1]) = [0,1]

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 (-\infty, a), 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.

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