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Is every region a domain? Am I correct that I understood the definition of domain to be an open, connected set? Does every region have to be a domain? For example: $|z-1+i|\le 3$ is a region if I've understood it correctly, but this is not open and therefore not a domain. Why would this be a region?

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closed as not a real question by Michael Lugo, Pete L. Clark, S. Carnahan, Anton Geraschenko Jan 18 '10 at 7:27

It's difficult to tell what is being asked here. This question is ambiguous, vague, incomplete, overly broad, or rhetorical and cannot be reasonably answered in its current form. For help clarifying this question so that it can be reopened, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.

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I am not aware of any technical meaning that "region" has in complex variable theory: so far as I know, a "region of the plane" is just a (perhaps not too pathological) subset of the plane. Do you have a particular text or article in mind? –  Pete L. Clark Jan 16 '10 at 21:19
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The definition I have read of region is that it is a connected open set in the complex plane. This is matter of your choice of complex analysis book. I do not think you should spend much time on this matter. –  Anweshi Jan 16 '10 at 21:20
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For instance my reference was Rudin, Real and complex analysis, in which he defines region as above, and for him a domain is the domain of a function(as opposed to co-domain). I suggest that this question be closed because it is too vague. –  Anweshi Jan 16 '10 at 21:22
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I have voted to close this question. My reasoning: it was a small question about terminology which deserved an answer, and it now has two quite satisfactory answers. I don't think it's worth someone else's time to answer it. (Is this a good reason to close? Let me know.) –  Pete L. Clark Jan 17 '10 at 6:40
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@Pete: I don't quite agree with your reason for closing. I'm voting to close because it's not clear what the question is. That is, somebody has to guess at some hidden information before answering. Is Anonymous asking for the definition of a region? If yes, this is absolutely the wrong question to ask. If no, then that's a crucial bit of missing information. If the question is, "I'm reading X and want to know the difference between the terms region and domain," then X is a very important bit of missing information. –  Anton Geraschenko Jan 18 '10 at 7:26

1 Answer 1

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.

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@Anonymous: This is equivalent to saying that a region is a domain plus some of its accumulation points. –  Harry Gindi Jan 16 '10 at 22:53
    
Sometimes "domain" is used in analysis to mean an open set which is the interior of its closure. For example, a punctured disk in not a domain in this sense. –  Ben McKay Oct 31 at 13:55

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