Definition given -
Region: Open set with none, some, or all of its boundary points.
This seems quite unimportant... and it seems that almost all sets are regions (I can only think of regions, I can't think of any example that isn't.).
It feels like it I'm given a set that is, neither open or closed, it could always be decomposed by taking away the boundary, meaning it is an open set with some of its boundary points.
This is why I think the following set is a region:
$$S = \{z=x+iy \in \mathbb{C}:x\geq 0, y>0\}.$$
If it is a region, what would be an example for a non-region?
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$\begingroup$[Too long to be a comment and this is not intended for answering your question directly.]
There is no general definition for this term and one should refer to the context regarding its precise meaning.
As Terry Tao points out in one of his lecture notes on complex analysis:
The notion of a non-empty open connected subset ${U}$ of the complex plane comes up so frequently in complex analysis that many texts assign a special term to this notion; for instance, Stein-Shakarchi refers to such sets as regions, and in other texts they may be called domains.
[Added:] The definition you gave "Open set with none, some, or all of its boundary points" is seldom used. To give an example which is not a region under this definition is rather easy: just consider a set with only a single point on the complex plane.
$\endgroup$ 5 $\begingroup$Consider the set of complex numbers: $\{ x+iy : x = y \}$ (the line $y=x$ in the 2D plane). This set cannot be expressed as an open set with some of its boundary points, since it has no interior points. It is therefore not a region according to your definition.
More generally, by your definition, any (non-empty) set with no interior points is not a region.
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