Locally Euclidean and Topological Manifolds

In summary, the discussion was about the definition of an n-manifold M, which includes the conditions of being Hausdorff, locally Euclidean of dimension n, and having a countable basis of open sets. The individual had a question about the second condition, which means that there exists a neighborhood around any point of M that is homeomorphic to a piece of the n-dimensional Euclidean space. They were confused because they thought the graph of the absolute value function was not Euclidean at the origin, but it was clarified that it is indeed a topological manifold, although not smooth as an embedded submanifold of R^2 due to the cusp at x=0.
  • #1
elarson89
20
0
Hello,

I am learning about manifolds but I am not understanding part of the definition. This is what I'm looking at for defining the n-manifold M.

(i) M is Hausdorff
(ii) M is locally Euclidean of dimension n, and
(iii) M has a countable basis of open sets

I have a problem with (ii). To the best of my knowledge this means there exists a neighborhood around an arbitrary point of M which is homeomorphic to a piece of [tex]\Re^n[/tex]. My confusion is that I had thought that the graph of the absolute value function was not euclidean at the origin. However it is clear to me that it is naturally homeomorphic to the real line, making this euclidean, and thus a manifold.

Please advise
 
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  • #2
The graph of the absolute function is indeed a topological manifold.

(But it is not smooth as an embedded submanifold of R^2 because of the cusp at x=0.)
 

1. What is a locally Euclidean manifold?

A locally Euclidean manifold is a type of mathematical space that is locally similar to Euclidean space, meaning that at any point on the manifold, there exists a neighborhood that is topologically equivalent to a Euclidean space of a certain dimension. In other words, the space appears flat and smooth at a small scale, but may have a more complex overall shape.

2. What is a topological manifold?

A topological manifold is a type of mathematical space that is locally similar to a Euclidean space, but may have a more complex overall shape. Unlike a Euclidean space, a topological manifold may have holes, self-intersections, or other topological features that make it different from a simple, flat space.

3. What are some examples of locally Euclidean and topological manifolds?

Examples of locally Euclidean and topological manifolds include spheres, tori, and the surface of a donut (also known as a torus). These manifolds can have complex shapes and topological features, but are locally similar to a Euclidean space at any given point.

4. What is the importance of studying locally Euclidean and topological manifolds?

Studying locally Euclidean and topological manifolds is important in many areas of mathematics and science, including topology, geometry, and physics. These spaces provide a way to understand and describe complex shapes and topological features that may not be easily visualized in a Euclidean space. They also have many practical applications, such as in computer graphics and data analysis.

5. How are locally Euclidean and topological manifolds different from other types of manifolds?

Locally Euclidean and topological manifolds are different from other types of manifolds, such as differentiable manifolds, in that they do not necessarily have a smooth or continuous structure. While differentiable manifolds require a differentiable structure, locally Euclidean and topological manifolds can have more general topological structures. Additionally, differentiable manifolds are often studied in the context of calculus and differential equations, while locally Euclidean and topological manifolds have broader applications in mathematics and science.

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