Boundary (Of A Manifold)

The subset of the closure $ \overline{ {M ^ {n} }}\; $ of an (open) $ n $- dimensional real manifold $ M ^ {n} $ for which a neighbourhood of each point is homeomorphic to some domain $ W ^ {n} $ in the closed half-space of $ \mathbf R ^ {n} $, the domain being open in $ \mathbf R _ {+} ^ {n} $( but not in $ \mathbf R ^ {n} $). A point $ a \in \overline{ {M ^ {n} }}\; $ corresponding to a boundary point of $ W ^ {n} \subset \mathbf R _ {+} ^ {n} $, i.e. to an intersection point of $ \overline{ {W ^ {n} }}\; $ with the boundary of $ \mathbf R _ {+} ^ {n} $, is called a boundary point of $ M ^ {n} $. A manifold having boundary points is known as a manifold with boundary. A compact manifold without boundary is known as a closed manifold. The set of all boundary points of $ M ^ {n} $ is an $ (n - 1) $- dimensional manifold without boundary.

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