- #1

Delong66

- 4

- 0

Thank you!

You are using an out of date browser. It may not display this or other websites correctly.

You should upgrade or use an alternative browser.

You should upgrade or use an alternative browser.

- I
- Thread starter Delong66
- Start date

In summary, the set D, which is a subset of R^2 with subspace topology, is not a manifold. It does not satisfy the requirement of having a neighborhood homeomorphic to an open subset of Euclidean space for every point. Additionally, removing the origin would result in 4 disconnected components, making it different from any surface or 1-manifold. However, if the origin is removed, it can be considered a topological manifold with 4 connected components, each being a 1-manifold and globally homeomorphic to the Reals.

- #1

Delong66

- 4

- 0

Thank you!

Physics news on Phys.org

- #2

- 4,132

- 1,733

- #3

WWGD

Science Advisor

Gold Member

- 7,339

- 11,292

- #4

jbergman

- 412

- 171

If you remove the origin, I believe it is a topological manifold.WWGD said:

- #5

WWGD

Science Advisor

Gold Member

- 7,339

- 11,292

Indeed, 4 lines, each a 1-manifold, globally homeomorphic to the Reals. A manifold with 4 connected components.jbergman said:If you remove the origin, I believe it is a topological manifold.

A differentiable manifold is a mathematical concept used to describe spaces that are locally similar to Euclidean spaces, but may have more complex global structures. It is a generalization of the concept of a smooth curve or surface in traditional geometry.

This refers to the dimensionality of the manifold. A 1-dimensional differentiable manifold is a curve, while a 2-dimensional differentiable manifold is a surface. In general, a d-dimensional differentiable manifold is a space that locally resembles d-dimensional Euclidean space.

We can determine if the coordinate axes are a differentiable manifold by checking if they satisfy the definition of a differentiable manifold. This includes being locally homeomorphic to Euclidean space, having a smooth structure, and being Hausdorff and second-countable.

Knowing if the coordinate axes are a differentiable manifold is important in many areas of science and mathematics, such as differential geometry, topology, and physics. It allows us to study and understand the properties of these spaces and apply them to real-world problems.

Differentiable manifolds have many applications in various fields, including computer graphics, robotics, image processing, and machine learning. They are also used in physics to describe the geometry of spacetime in general relativity and in the study of dynamical systems and chaos theory.

- Replies
- 3

- Views
- 839

- Replies
- 36

- Views
- 1K

- Replies
- 44

- Views
- 2K

- Replies
- 10

- Views
- 1K

- Replies
- 7

- Views
- 3K

- Replies
- 4

- Views
- 2K

- Replies
- 4

- Views
- 697

- Replies
- 10

- Views
- 3K

- Replies
- 6

- Views
- 832

- Replies
- 13

- Views
- 2K

Share: