Should continuum be an open subset?

I recent read the book of Prof. J. Tinsley Oden "A short course on nonlinear continuum mechanics"

 http://users.ices.utexas.edu/~arbogast/cam397/oden0908.pdf

 which declares that reference configuration is an open, bounded, connected subset  of R3 with a smooth boundary (in Page 1).

It is strange to me beacause that means we cannot consider the boundary of the deformable bodies in its following discussion.

   What is that declaration mean? And shouldn't continuum be an open subset?