Problem 3 - Calculus On Manifolds: A Modern Approach To Classical Theorems Of Advanced Calculus

Question

5-3. (a) Let \(A \subset \mathbf{R}^{n}\) be an open set such that boundary \(A\) is an \((n-1)\) dimensional manifold. Show that \(N=A \cup\) boundary \(A\) is an \(n\)-dimensional manifold-with-boundary. (It is well to bear in mind the following example: if \(A=\left\\{x \in \mathbf{R}^{n}:|x|<1\right.\) or \(\left.1<|x|<2\right\\}\) then \(N=A \cup\) boundary \(A\) is a manifold-with- boundary, but \(\partial N \neq\) boundary \(A .)\) (b) Prove a similar assertion for an open subset of an \(n\)-dimensional manifold.

Step-by-Step Solution

Answer

When merging an open set and its \( (n-1) \)-dimensional boundary, the result is an \( n \)-dimensional manifold-with-boundary in both \( \textbf{R}^n \) and general \( n \)-dimensional manifolds.

Step 1 - Understand the Problem

We need to show that when an open set in \(\textbf{R}^n\) with an \((n-1)\)-dimensional manifold as its boundary is combined with its boundary, the resulting set is an \(n\)-dimensional manifold-with-boundary.

Step 2 - Definition of Manifold-with-Boundary

Recall the definition of a manifold-with-boundary: a space is an \(n\)-dimensional manifold-with-boundary if each point has a neighborhood homeomorphic to either \(\textbf{R}^n\) or the upper half-space \(\textbf{H}^n\).

Step 3 - Open Set in \(\textbf{R}^n\)

Let \(A\) be an open set in \(\textbf{R}^n\) and assume its boundary, \(\text{boundary } A\), is an \((n-1)\)-dimensional manifold.

Step 4 - Neighborhoods in Set A

For any point \(p\) in \(A\), there exists a neighborhood around \(p\) homeomorphic to \(\textbf{R}^n\) because \(A\) is an open set in \(\textbf{R}^n\).

Step 5 - Neighborhoods on the Boundary

For any point \(q\) on the boundary of \(A\), there exists a neighborhood homeomorphic to the upper half-space \(\textbf{H}^n\) because the boundary is an \((n-1)\)-dimensional manifold, fitting the definition of a manifold-with-boundary. Thus, it satisfies the local homeomorphism condition.

Step 6 - Merging A and its Boundary

Combine \(A\) and \(\text{boundary } A\) to form \(N = A \cup \text{boundary } A\). Since every point in \(N\) has a neighborhood homeomorphic to either \(\textbf{R}^n\) (if the point is in \(A\)) or \(\textbf{H}^n\) (if the point is on the boundary), \(N\) is indeed an \(n\)-dimensional manifold-with-boundary.

Step 7 - Analyzing \(\textbf{R}^n\) Manifolds

Now let \(M\) be an \(n\)-dimensional manifold and let \(B \subset M\) be an open set such that its boundary \(\text{boundary } B\) is an \((n-1)\)-dimensional manifold in \(M\).

Step 8 - Applying Previous Steps to General Manifolds

The same argument holds: neighborhoods on internal points correspond to \(\textbf{R}^n\) and neighborhoods on the boundary points correspond to the upper half-space \(\textbf{H}^n\), ensuring \(B \cup \text{boundary } B\) is an \(n\)-dimensional manifold-with-boundary.

Key Concepts

Open Set

In the realm of topology, an open set is a fundamental concept. An open set within a space is a set where, for any point within the set, there is a neighborhood around that point entirely contained within the set. In simpler terms, you can move a little bit in any direction from any point in the set without leaving it.
For instance, if you consider an open interval \((a, b)\) in \(\textbf{R}^1\), any point \(x\) within this interval has a small interval around it that still stays within \((a, b)\). More generally, this applies to higher dimensions like \(\textbf{R}^n\), where every point in an open set has a small open ball around it that lies within the set.
It forms an essential building block for defining more complex structures like manifolds.

n-Dimensional Manifold

An \(n\)-dimensional manifold is a space that, locally, resembles \(\textbf{R}^n\). This means that around every point in the manifold, there is a neighborhood homeomorphic (a fancy term for a mapping that preserves properties like shape) to an open subset of \(\textbf{R}^n\). They are the higher-dimensional analogs to curves (1-D) and surfaces (2-D).
For example, the surface of a sphere is a 2-dimensional manifold. Locally, every point on the surface looks like a flat plane (open subset of \(\textbf{R}^2\)). Similarly, in higher dimensions, these local properties hold, making them a central topic in advanced geometry and physics.

Boundary of a Manifold

The boundary of a manifold is akin to the edge or the limits of the manifold. Specifically, in an \(n\)-dimensional manifold, the boundary is an \((n-1)\)-dimensional subset. Take, for instance, a disk in \(\textbf{R}^2\); the boundary here is the circle enclosing the disk.
Formally, for a set to be an \(n\)-dimensional manifold-with-boundary, each point in the set must either have a neighborhood homeomorphic to \(\textbf{R}^n\) (if it is an interior point) or to the upper half-space \(\textbf{H}^n\) (if it is on the boundary).
In this exercise, the set \(N = A \cup \text{boundary } A\) includes both the open set \(A\) and its boundary, ensuring \(N\) retains the characteristics of an \(n\)-dimensional manifold with boundary properties.

Local Homeomorphism

A local homeomorphism is a mapping between two topological spaces that is a homeomorphism when restricted to a small neighborhood around each point in its domain. What this means is, looking locally, the structures of the spaces are preserved under the mapping.
For instance, consider the mapping from a small neighborhood of a point on an open set \(A\) in \(\textbf{R}^n\) to an open set in \(\textbf{R}^n\). The mapping looks like stretching or shrinking within the neighborhood but does not tear or glue parts together. This ensures that the local topological properties, like continuity, are maintained.
Local homeomorphisms are crucial in defining manifolds, as they ensure that each small piece of the manifold behaves nicely in terms of standard Euclidean spaces.

Upper Half-Space

The upper half-space is a concept often used in manifold theory. It is the set of all points in \(\textbf{R}^n\) for which the last coordinate is non-negative. Mathematically, this is written as \(\textbf{H}^n = \{ (x_1, x_2, ..., x_n) \in \textbf{R}^n | x_n \geq 0 \}\).
This space models the idea of points lying either on or above a boundary, such as a plane. It's crucial in the context of manifolds-with-boundary, as it serves as the local model for neighborhoods containing boundary points.