Atlas of Manifold: Is it Countable?

In summary, the conversation discusses the question of whether an atlas, which is used to describe the structure of a manifold, is a countable set. It is mentioned that every manifold is second countable, meaning it has a countable basis. However, it is unclear if every element of the basis can fit inside a chart domain, which would determine if the atlas is countable. Additionally, the conversation touches on the fact that manifolds are not necessarily second countable, depending on the author and the specific context.
  • #1
Pietjuh
76
0
I was thinking about something yesterday and I couldn't quite figure it out. It's about the question if an atlas is a countable set. Because we know that every manifold is second countable, so it has a countable basis. But does every element of the basis fit inside a chart domain? If that's the case then the atlas is countable. But I'm not sure that's the case :)
 
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  • #2
Each n-sphere S^n is covered by an atlas that has 2 members.

Do you mean maximal atlas?

Regards,
George
 
  • #3
Not only that but manifolds aren't necessarily second countable. It depends on the author. If you start with a set M, and put a complete smooth atlas on it (so I'm talking about differentiable manifolds in this context), then the charts form a basis for the topology on the M and that topology isn't necessarily second countable.


kevin
 

FAQ: Atlas of Manifold: Is it Countable?

1. What is the Atlas of Manifold?

The Atlas of Manifold is a mathematical concept that represents the collection of all possible charts or coordinate systems on a manifold. It helps in understanding the structure and properties of the manifold.

2. What is a manifold?

A manifold is a mathematical object that can be described locally by coordinates and is similar to the concept of a surface in three-dimensional space. It can be visualized as a higher-dimensional space that is curved or twisted in some way.

3. What does it mean for a manifold to be countable?

A countable manifold is one that has a finite or countably infinite number of charts or coordinate systems. This means that the manifold can be covered by a finite or countably infinite number of smaller, simpler pieces.

4. How is countability related to the Atlas of Manifold?

The countability of a manifold is directly related to the Atlas of Manifold, as the Atlas represents all possible coordinate systems on the manifold. If the manifold is countable, then the Atlas will also be countable.

5. Why is the countability of a manifold important?

The countability of a manifold is important in understanding its structure and properties. It allows for a better understanding of the manifold's topology and can help in solving mathematical problems related to it.

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