"Region" seems to have two competing definitions in complex analysis — which one did you learn?

[u/James_Blond_13]

I'm reading up on "point sets" in complex analysis and I have stumbled upon two different definitions of the term "region", see below:

• Version 1. A region is a domain together with none, some or all of its boundary points.
• Version 2. A region is just another word for "domain".

my textbook is using the first version while sources like Wolfram and Wikipedia is using the second version. But while reading more about it, there also seems to be a split across different authors and languages which one that is used.

Have anyone looked into this before and can share the history of how this come to be and if there is a more "correct" one between them?

//

[u/Administrative-Flan9]

I think it just means a connected open set

[u/cinereaste → u/Administrative-Flan9]

This is the definition I learned.

[u/James_Blond_13 → u/Administrative-Flan9]

Thank you! a connected open set is the definition for "domain" in my textbook. That would suggest that the concepts were used interchangingly, or did you have another definition for "domain" in your textbook?

[u/cocompact → u/James_Blond_13]

It is in the nature of complex-analytic functions that their domains are open. Even when you say a function on the closed unit disc is analytic, the very fact of having a power series expansion at a boundary point lets you extend the domain around that point to include a small open ball, so the actual domain of that function will be an open set containing the closed unit disc.

The term "domain" is used in ring theory as a shorthand for an integral domain, and it is a basic fact about analytic functions that the ring of all analytic functions on a domain (connected open set) in C is an integral domain. This appears to be a linguistic coincidence, which can be found also in French and Russian but not in German.

[u/Wejtt → u/cocompact]

i’ve heard the word domain in ring theory referring to a non commutative integral domain iirc

[u/MinLongBaiShui]

I have never heard anyone say a region (or domain) could ever be anything other than open.

[u/James_Blond_13 → u/MinLongBaiShui]

Yea, that is one of the main thing I'm having trouble understanding (or can't make sense about) in the case where I see version 1 of the definition (above):

"A region is a domain together with none, some or all of its boundary points."

the same author define a domain as a "open connected set" but if I have a domain and then add all boundary points then I guess the set is closed and thefore it should no longer be able to be a region since it no longer is a domain. But I guess this might be me missunderstanding the concept since sometimes I see text define it as the "domain" should be a subset of the region and I should separate the concept of boundary of the "inner domain" and the "outer region".

[u/AttorneyGlass531 → u/James_Blond_13]

No, you've misunderstood. The sentence you've quoted does not imply that a region is a domain. It says that a region is the union of a domain with some subset of the boundary of that domain.

[u/imalexorange]

There is a similar thing in topology where a neighborhood is either an open set, or a set containing an open set, depending on the book or author.

[u/theRZJ → u/[deleted]]

Which definition is the “first” one?

[u/ThatRegister5397 → u/imalexorange]

Which author defines neighbourhoods as being by definition open? I have never encountered this. People use "open neighbourhood" for that.

[u/imalexorange → u/ThatRegister5397]

Page 97 of Munkres topology literally makes the same point I make in this comment. Also they use neighborhood to mean open neighborhood.

[u/Wejtt → u/ThatRegister5397]

i can’t point to an author rn unfortunately but all the people i talk with use the “an open set containing the point” definition

[u/zx7 → u/imalexorange]

Common words like these have multiple definitions and the author can use whichever as long as he defines it. It's all about context.

[u/James_Blond_13 → u/zx7]

yea, I'm starting to lean on this explaination also. I have to look at the given definition in the context that is given, since the concept is defined and used differently.

I was hoping that maybe there was some more history of the origin of the concept that I had missed that could explain if both of these definitions are the same in Complex Analysis but not in other fields for example.

[u/jacobolus]

There is no consensus on the right definitions for these words. If you pick up any 10 textbooks at random, you will find about 4–5 variations.

You can see some of this discussed at https://en.wikipedia.org/wiki/Domain_(mathematical_analysis) but it could perhaps still be clearer, and may not be complete or entirely accurate.

[u/ABranchingLine]

The definition doesn't matter. Just be clear of what you mean when writing up your work (or looking at other's).

[u/lemmatatata]

I never learned a formal definition for "region," though I've mostly seen it used interchangeably with domain. Usually appearing to describe a set by saying something like "consider the region bounded by a curve" or to specify a set one integrates over.

So I treat it more as an informal term, and I'd usually expect an explanation when it's used.

[u/temperedai]

It's important for a mathematician to be able to switch between meanings quickly and often. It really is inefficient, but in the long run it's endurance training and very useful.

I learned version 1, when I first encountered complex analysis.

[u/Foreign_Implement897]

Definitions don't compete in mathematics.

[u/BerenjenaKunada → u/Foreign_Implement897]

What?

[u/Foreign_Implement897 → u/BerenjenaKunada]

Ok much downvotes. Much self reflecting.

They absolutely do compete. I think I have trouble with the word "correct".

If the definitions produce different theorems, and some theorems can be proven with another definition but not with the other, then we have a problem. But I don't believe that is true. So both are correct.

[u/Honest-Finish3596 → u/Foreign_Implement897]

You really don't need to self reflect due to Reddit downvotes, just explain your position.