Maybe I should make some standardizing comments.
First, in mathematics the
k-dimensional sphere always refers to the k-dimensional object {x: |x| = r}. { x : |x| <= r } is called the closed ball and { x : |x| < r } the open ball. So the k-dimensional sphere, or short S<sub>k</sub>, is the boundary of the k+1 dimensional closed ball. Mostly used in topology the radius r does not matter and the sphere is up to isomorphism. It makes sense to use these conventions also throughout this forum.
Next the term
closed is ambiguous. If we talk of a
closed surface/manifold, it means that it has no boundary. If we talk of a closed (general) set, it means that the complement is open, which means in turn that each point has an environment that also belongs to the set (where environment is a base topological term).
To make it more difficult the term
boundary is differently (i.e. more specific)
defined for manifolds than for general sets (which was PWrong's definition).