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kostas230

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In summary, the conversation discusses using the stereographic projection to prove that the closed unit ball is a manifold with boundary. The formula for the stereographic projection and its inverse is requested, and its continuity is verified. The part of the n-sphere described by σ^−1(B) is also discussed, as well as the composition of σ^−1 with projections to create charts and their continuity.

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kostas230

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Anyway, consider the closed unit ball ##B## of ##\mathbb{R}^n##. Consider the ##n##-sphere ##S^n##. Consider the stereographic projection ##\sigma:S^n\rightarrow \mathbb{R}^n##. Can you give a formula for the stereographic projection and its inverse? Can you verify it is continuous.

Then, what can you say about ##\sigma^{-1}(B)##. It is a part of the ##n##-sphere ##S^n##, but can you describe which part it is?

Charts can then be found by composing ##\sigma^{-1}## with the projections ##\pi:\mathbb{R}^{n+1}\rightarrow \mathbb{R}^n## which leave some coordinate. Can you show that ##\pi\circ \sigma^{-1}## are continuous? Can you describe these things explicitely? Can you tell what the right codomain is? Etc.

- #3

kostas230

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Thanks! I was stuck in σ^−1(B) for no reason. xD

A closed ball is a set of points in a metric space that are all within a certain distance, called the radius, of a given point. It includes all points on the boundary of the ball as well as the interior points.

A manifold with boundary is a topological space that is locally homeomorphic to Euclidean space, but may have a boundary that is not homeomorphic to a point. In other words, it is a space that is locally similar to a flat surface, but may have a curved or irregular boundary.

A closed ball can be seen as a simple example of a manifold with boundary. It is a topological space that is locally homeomorphic to a closed interval, with the boundary point being the center of the ball.

The fact that a closed ball is a manifold with boundary has important implications in mathematics and physics. It allows for the use of techniques and concepts from manifold theory in the study of closed balls, making it a powerful tool for understanding and solving problems in various fields.

The boundary of a closed ball is a single point, while the boundary of a manifold without boundary is empty. This means that while a closed ball has a well-defined boundary point, a manifold without boundary does not, and its boundary is considered to be nonexistent.

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