What is the "boundary" in Stoke's theorem?

What is the "boundary" in Stoke's theorem?

The classical Stokes' theorem "relates the surface integral of the curl of
a vector field over a surface Ó in Euclidean three-space to the line
integral of the vector field over its boundary". As far as I can tell,
this definition assumes that the boundary of Ó is a closed path.
However, the topological boundary of a surface in R^3 is not always a path
-- it's typically the entire surface. For example, every point of the unit
disk in R^3 is a boundary point (in the standard topology of R^3).
How exactly is the "boundary of Ó" defined? Is it the topological boundary
of Ó in the subspace topology of Ó?