The closed unit ball \( B \) in \( \mathbb{R}^n \) is indeed a manifold with boundary, as established through the use of stereographic projection \( \sigma: S^n \rightarrow \mathbb{R}^n \). The stereographic projection can be defined explicitly, and its inverse \( \sigma^{-1} \) maps points from \( \mathbb{R}^n \) back to the \( n \)-sphere \( S^n \). The continuity of both \( \sigma \) and \( \sigma^{-1} \) is crucial for demonstrating that \( \sigma^{-1}(B) \) corresponds to a specific part of the \( n \)-sphere, which can be further analyzed through the composition with projection maps \( \pi: \mathbb{R}^{n+1} \rightarrow \mathbb{R}^n \).
PREREQUISITESGraduate mathematics students, particularly those studying differential geometry and topology, as well as researchers interested in manifold theory and its applications.