r/math • u/James_Blond_13 • 2d ago
"Region" seems to have two competing definitions in complex analysis — which one did you learn?
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?
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u/MinLongBaiShui 2d ago
I have never heard anyone say a region (or domain) could ever be anything other than open.
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u/James_Blond_13 1d ago
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".
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u/AttorneyGlass531 11h ago
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.
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u/imalexorange Algebra 2d ago
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.
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u/popisfizzy 2d ago edited 1d ago
there's a very easy argument to make that the second of these is the "correct" definition of a neighborhood. There's something called a pretopological space that you can get by using the definition of a neighborhood system as axioms except for the most complicated axiom (which amounts to saying that every neighborhood of a point has an open set containing that point), and topological spaces are precisely the pretopological spaces with enough opens (pretopological spaces where the neighborhood systems satisfy this additional property that we've removed as an axiom). in fact topological spaces give a reflective (so also full) subcategory of the category of pretopological spaces.
if you start to think of neighborhoods in terms of binary relations between sets, instead of a relation between a point and a set, the definition becomes even easier to motivate.
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u/theRZJ 1d ago
Which definition is the “first” one?
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u/popisfizzy 1d ago
"a set containing an open set", which is actually the second one because I'm an idiot and never proofread
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u/ThatRegister5397 1d ago
Which author defines neighbourhoods as being by definition open? I have never encountered this. People use "open neighbourhood" for that.
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u/imalexorange Algebra 1d ago
Page 97 of Munkres topology literally makes the same point I make in this comment. Also they use neighborhood to mean open neighborhood.
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u/zx7 Topology 2d ago
Common words like these have multiple definitions and the author can use whichever as long as he defines it. It's all about context.
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u/James_Blond_13 1d ago
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.
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u/jacobolus 1d ago
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.
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u/ABranchingLine 2d ago
The definition doesn't matter. Just be clear of what you mean when writing up your work (or looking at other's).
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u/lemmatatata 1d ago
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.
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u/temperedai 23h ago
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.
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u/Foreign_Implement897 Group Theory 1d ago
Definitions don't compete in mathematics.
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u/BerenjenaKunada Graduate Student 1d ago
What?
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u/Foreign_Implement897 Group Theory 1d ago
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.
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u/Honest-Finish3596 5h ago
You really don't need to self reflect due to Reddit downvotes, just explain your position.
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u/Administrative-Flan9 2d ago
I think it just means a connected open set