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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?

Twenty-six colours
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    The term is a somewhat vague term. I would say its meaning depends on context. So no need to focus too much on the definition of the term. To put it in another way, once you can grasp the concept of set then you won't be bothered by this term :). – Yes Jul 25 '17 at 14:05
  • Ah I see; thank you. I think I do grasp the concept of a set but maybe not, but unsure how it would relate to the region definition – Twenty-six colours Jul 25 '17 at 14:16

2 Answers2

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[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.

  • My course distinguishes the two, defining a region as above and the domain as $U$ in that note – Twenty-six colours Jul 25 '17 at 14:17
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    Well, yes, there is no general definition and its meaning depends on the context. –  Jul 25 '17 at 14:20
  • This doesn't really answer the question, which was asking more about the specific definition given and what kind of sets fall under it. – Sambo Jul 25 '17 at 14:32
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    Fair enough. Example added. For me, this is nothing but a notion for convenience of doing analysis. I doubt that people in practice would really care about classifying which sets are "region" and which are not. –  Jul 25 '17 at 14:52
  • True, but I think the point of the question was just to get some intuition about the definition given. – Sambo Jul 25 '17 at 15:06
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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.

Sambo
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    Does the empty set have an interior? –  Jul 25 '17 at 14:46
  • @Jack Any set contains its interior, so the interior of the empty set is empty. I guess since the empty set is technically open, and hence technically a region by this definition, I can add this condition. – Sambo Jul 25 '17 at 15:05