cv.complex variables - Region and domains?

Standard definitions in geometric complex analysis are as follows:

A domain is a nonempty open connected set (just as in analysis in general).

A region is a set whose interior is a domain and which is contained in the closure of its interior.

For example the open unit disk and none, part, or all of its boundary (the unit circle).

The closed unit disk together with the interval $[1,2]$ on the real axis is not a region.