On General Definition of Sobolev Spaces

Question

This might be trivial, but why a Sobolev space is generally defined with respect to an open set and not to a closed set or more generally? Is it related to the differentiability concept?

P.S.: The point I've said about the definition of a Sobolev space with respect to an open set can be supported by the following references:

1. Tartar, 2007. An Introduction to Sobolev Spaces and Interpolation Spaces.

2. https://math.aalto.fi/~jkkinnun/files/sobolev_spaces.pdf

3. https://en.wikipedia.org/wiki/Sobolev_space#Extension_by_zero

And one seeming exception can be seen here: https://www.ljll.math.upmc.fr/frey/cours/UdC/ma691/ma691_ch2.pdf (or maybe it is not well-defined)

Thanks