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Mathematics for Physics

137 Boundary

A boundary is a closed set of elements associated with a subset that is used to define the limit of a neighborhood where a function of a variable has a constant scalar value.

definition [d] (Boundary = Topological Boundary) From Nash and Sen: the boundary of a set \(U'\), written \(b(U')\), is the complement of the interior of \(U'\) in the closure of \(U'\):

  • \(b(U') = \overline{U'} \setminus U'^{\circ}\) .

where

  • \(U'\) is a subset of a topological space.
  • \(U'^{\circ}\) is the interior of \(U'\).
  • \(\overline{U'}\) is the closure of \(U'\).
  • \(b(U')\) is the boundary of \(U'\).

Note:

  • Nash and Sen: if \(U = [a,b)\) on the real line \(\mathbb{R}\), then \(U^{\circ} = (a,b)\) and \(\overline{U} = [a,b]\), so \(b(U) = \{a,b\}\).
  • Nash and Sen: the sets \((a,b)\), \([a,b]\), \([a,b)\), and \((a,b]\) all have the same boundary \(\{a,b\}\).
  • Nash and Sen: \(U^{\circ} = U\) if and only if \(U\) is open.

definition [d] (Boundary = Manifold Boundary) From Lee: an \(n\)-dimensional manifold with boundary is a second countable Hausdorff space in which every point has a neighborhood homeomorphic either to an open subset of \(\mathbb{R}^{n}\) or to an open subset of the closed upper half-space

  • \(\mathbb{H}^{n} = \{ (x_{1},\ldots,x_{n}) \in \mathbb{R}^{n} : x_{n} \ge 0 \}\) .

If \(M\) is an \(n\)-manifold with boundary, a point \(p \in M\) is called an interior point of \(M\) if it is in the domain of an interior chart, and a boundary point of \(M\) if it is in the domain of a boundary chart that takes \(p\) to \(\partial \mathbb{H}^{n}\). The boundary of \(M\), denoted \(\partial M\), is the set of all its boundary points, and its interior, denoted \(\operatorname{Int} M\), is the set of all its interior points. Every point of \(M\) is either an interior point or a boundary point.

where

  • \(M\) is an \(n\)-manifold with boundary.
  • \(\mathbb{H}^{n}\) is the closed upper half-space in \(\mathbb{R}^{n}\).
  • \(\partial M\) is the manifold boundary of \(M\).
  • \(\operatorname{Int} M\) is the manifold interior of \(M\).

Note:

  • Lee: despite the terminology, a manifold with boundary is not necessarily a manifold.

137.1 Elementary Example

137.1.1 Simple

The boundary is closure minus interior. For a half-open interval, the boundary is the two endpoints.

\[ U = [a,b) \]

\[ U^{\circ} = (a,b),\quad \overline{U} = [a,b] \]

\[ b(U) = \overline{U} \setminus U^{\circ} = \{ a,\ b \} \]

where

  • \(U\) is the subset.
  • \(U^{\circ}\) is the interior of \(U\).
  • \(\overline{U}\) is the closure of \(U\).
  • \(b(U)\) is the boundary of \(U\).

137.1.2 General

Open, closed, and half-open intervals with the same endpoints share the same boundary.

\[ (a,b),\ [a,b],\ [a,b),\ (a,b] \]

\[ b(U) = \{ a,\ b \} \]

where

  • \(a\) and \(b\) are real numbers with \(a < b\).
  • \(b(U)\) is the common boundary set for each of those intervals \(U\).

137.2 References

  1. Nash, C., & Sen, S. Topology and Geometry for Physicists. Academic Press, 1983. — \(b(U') = \overline{U'} \setminus U'^{\circ}\); example \(U=[a,b)\) gives \(b(U)=\{a,b\}\).
  2. Lee, J. M. Introduction to Topological Manifolds. Springer. — \(\mathbb{H}^{n}\); manifold with boundary; interior point; boundary point; \(\partial M\); \(\operatorname{Int} M\).