Let S be a geometric surface.
Definition: [Boundary of a surface] A surface with a boundary is a surface along with boundary points. A boundary point $P$ is a point such that there exists an open set $U$ in $R^3$ so that $P ∈ U ∩ S$ is homeomorphic to a half disk in. The set of all boundary points is called the boundary of $S$ and is denoted by $∂S$. Caution: Boundary of a subset of $R^3$ is not the same as the boundary of a surface in $R^3$.
This is the definition of the boundary of a surface. I am having trouble understanding why a sphere, or a torus doesn't have a boundary, according to the definition. Help would be appreciated :)