Topological Properties of the Closed Unit Ball

Oct 13, 2025 by pinkandtrash.com 18 views

Introduction

In the realm of differential geometry and topology, manifolds play a crucial role in understanding the properties of spaces that are locally Euclidean. A manifold with boundary is a topological space that is locally Euclidean, but with the added feature of having a boundary. In this article, we will explore the concept of a closed unit ball as a topological $m$ -manifold with boundary $\mathbb{S}^m$ . This exercise is inspired by examples 1.4 and 1.31 of Lee's introduction to Smooth Manifolds, where the sphere is shown to be a smooth manifold.

What is a Manifold with Boundary?

A manifold with boundary is a topological space that is locally Euclidean, meaning that every point in the space has a neighborhood that is homeomorphic to an open subset of Euclidean space. However, unlike a manifold without boundary, a manifold with boundary has a boundary, which is a subset of the space that is defined as the set of points that are not in the interior of the space. In other words, the boundary of a manifold with boundary is the set of points that are "on the edge" of the space.

The Closed Unit Ball

The closed unit ball in $\mathbb{R}^m$ is the set of all points $x = (x_1, x_2, \ldots, x_m)$ such that $x_1^2 + x_2^2 + \ldots + x_m^2 \leq 1$ . This is a closed and bounded subset of $\mathbb{R}^m$ , and it is a fundamental object of study in geometry and topology.

Topological Properties of the Closed Unit Ball

To show that the closed unit ball is a topological $m$ -manifold with boundary $\mathbb{S}^m$ , we need to establish several topological properties of the space. First, we need to show that the closed unit ball is locally Euclidean. This means that every point in the space has a neighborhood that is homeomorphic to an open subset of Euclidean space.

Local Euclideanity

Let $x$ be a point in the closed unit ball. We need to show that there exists a neighborhood of $x$ that is homeomorphic to an open subset of Euclidean space. To do this, we can use the following argument. Let $r$ be a positive real number such that $r < 1 - \|x\|$ . Then the open ball $B(x, r)$ is a neighborhood of $x$ that is homeomorphic to an open subset of Euclidean space. Specifically, the map $\phi: B(x, r) \to B(0, r)$ defined by $\phi(y) = \frac{y - x}{r}$ is a homeomorphism.

Boundary of the Closed Unit Ball

The boundary of the closed unit ball is the set of points that are not in the interior of the space. In other words, the boundary is the set of points that are "on the edge" of the space. To show that the boundary of the closed unit ball is $\mathbb{S}^m$ , we need to establish that every point on the boundary is a limit point of the space.

Limit Points

Let $x$ be a point on the boundary of the closed unit ball. We need to show that $x$ is a limit point of the space. To do this, we can use the following argument. Let $r$ be a positive real number such that $r < 1 - \|x\|$ . Then the open ball $B(x, r)$ contains a point $y$ that is not in the interior of the space. Specifically, we can choose $y$ to be a point on the boundary of the closed unit ball that is closest to $x$ . Then $y$ is a limit point of the space, and therefore $x$ is also a limit point of the space.

Conclusion

In this article, we have shown that the closed unit ball is a topological $m$ -manifold with boundary $\mathbb{S}^m$ . We have established several topological properties of the space, including local Euclideanity and the existence of limit points on the boundary. This exercise is inspired by examples 1.4 and 1.31 of Lee's introduction to Smooth Manifolds, where the sphere is shown to be a smooth manifold.

References

Lee, J. M. (2013). Introduction to Smooth Manifolds. Springer.
Hirsch, M. W. (1976). Differential Topology. Springer.

Q: What is a manifold with boundary?

A: A manifold with boundary is a topological space that is locally Euclidean, but with the added feature of having a boundary. The boundary of a manifold with boundary is the set of points that are not in the interior of the space.

Q: What is the closed unit ball?

A: The closed unit ball in $\mathbb{R}^m$ is the set of all points $x = (x_1, x_2, \ldots, x_m)$ such that $x_1^2 + x_2^2 + \ldots + x_m^2 \leq 1$ . This is a closed and bounded subset of $\mathbb{R}^m$ .

Q: Why is the closed unit ball a topological $m$ -manifold with boundary $\mathbb{S}^m$ ?

A: The closed unit ball is a topological $m$ -manifold with boundary $\mathbb{S}^m$ because it satisfies several topological properties, including local Euclideanity and the existence of limit points on the boundary.

Q: What is local Euclideanity?

A: Local Euclideanity is the property of a space that every point in the space has a neighborhood that is homeomorphic to an open subset of Euclidean space.

Q: How do you show that the closed unit ball is locally Euclidean?

A: To show that the closed unit ball is locally Euclidean, we can use the following argument. Let $x$ be a point in the closed unit ball. We can choose a positive real number $r$ such that $r < 1 - \|x\|$ . Then the open ball $B(x, r)$ is a neighborhood of $x$ that is homeomorphic to an open subset of Euclidean space.

Q: What is the boundary of the closed unit ball?

A: The boundary of the closed unit ball is the set of points that are not in the interior of the space. In other words, the boundary is the set of points that are "on the edge" of the space.

Q: How do you show that the boundary of the closed unit ball is $\mathbb{S}^m$ ?

A: To show that the boundary of the closed unit ball is $\mathbb{S}^m$ , we need to establish that every point on the boundary is a limit point of the space. We can use the following argument. Let $x$ be a point on the boundary of the closed unit ball. We can choose a positive real number $r$ such that $r < 1 - \|x\|$ . Then the open ball $B(x, r)$ contains a point $y$ that is not in the interior of the space. Specifically, we can choose $y$ to be a point on the boundary of the closed unit ball that is closest to $x$ . Then $y$ is a limit point of the space, and therefore $x$ is also a limit point of the space.

Q: What are some common applications of manifolds with boundary?

A: Manifolds with boundary have many applications in mathematics and physics, including:

Differential geometry: Manifolds with boundary are used to study the properties of curves and surfaces in Euclidean space.
Topology: Manifolds with boundary are used to study the properties of spaces that are locally Euclidean.
Physics: Manifolds with boundary are used to model the behavior of physical systems, such as the motion of particles in a potential field.

Q: What are some common mistakes to avoid when working with manifolds with boundary?

A: Some common mistakes to avoid when working with manifolds with boundary include:

Failing to distinguish between the interior and boundary of a manifold.
Failing to check that a map is a homeomorphism.
Failing to check that a map preserves the boundary of a manifold.

Q: What are some resources for further reading on manifolds with boundary?

A: Some resources for further reading on manifolds with boundary include:

Lee, J. M. (2013). Introduction to Smooth Manifolds. Springer.
Hirsch, M. W. (1976). Differential Topology. Springer.
Spivak, M. (1999). A Comprehensive Introduction to Differential Geometry. Publish or Perish.