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I have two simple questions regarding the writing styles.

  1. Assume that I make a remark~$1.1$ and I define two sets say $A:=\text{something}$ and $B:=\text{something}$. My question is that I will use the sets $A$ and $B$ in many proofs in the paper. Should I say at the beginning of each proof, take $A$ and $B$ as in the remark~$1.1$? OR just write $A$ and $B$ and the reader should check what $A$ and $ B$ are in the paper?
  1. What is the best way to rewrite the following " Let $A$ and $B$ as in the Theorem~1 and $D$ and $E$ as in the remark~2. Let $\mathcal H$ be a family as in the Theorem~2 with $A.$ Also, let $\mathcal G$ be a family as in the Proposition~3."

Any help will be greatly appreciated.

FShrike
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00GB
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    For (2), what you have is more-or-less common practice. Having so many previous references at once may be a little off-putting to the reader ... especially if the definitions of $A$, $B$, $D$, $E$, etc, are buried in exposition. So, make an effort to have the previous definitions stand out as much as possible (say, as "displayed" equations). You could/should also advise the reader at the time that they'll be seeing these definitions again (often?). ... All that said, if this is for a journal or other publication, the editorial guidelines and/or staff may have specific recommendations. – Blue Aug 31 '21 at 20:29
  • @Blue, Assume all definitions are done in the correct way. Could you show another way to say this? – 00GB Aug 31 '21 at 20:36
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    "Could you show another way to say this?" Without specific context, it's difficult to advise. What you have is okay. It would probably help to slip in some brief reminders about what the symbols mean; eg, "Let $A$ and $B$ be the open sets used in the proof of Theorem 1, and let $C$ and $D$ be the pathological examples we described in Remark 2." Stuff like that. ... But the best thing to do is have someone proofread your actual work. – Blue Aug 31 '21 at 20:58
  • @Blue, Thank you so much. – 00GB Aug 31 '21 at 21:01
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    While some use of this "global notation" technique can be useful at times, I really recommend that you repeat as many hypotheses as possible in the statement of each result. Most readers will indeed be demoralized, to the point of putting the paper down, upon encountering a statement of the type quoted in 2. – Greg Martin Sep 01 '21 at 06:46
  • @GregMartin, Thank you, it is a good point. – 00GB Sep 01 '21 at 14:23

2 Answers2

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There is no rule, and no uniform "best" way to do this.

If you refer to an object often, you need not point the reader back to the definition each time. If you define it, use it once right away and then again much further on then the back reference might help.

In any case don't put a definition in a remark. Make it a definition.

Ethan Bolker
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  • Ethan Bolker, They are not really definition some sets that I will use many times and I need to put some notation for them. I do not think I can write them as definition – 00GB Aug 31 '21 at 20:17
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    The general point stands, if it is something the reader is expected to know/refer to frequently, the body of the text in a big stand out way would be more useful. Something like "Our sample sets will be A:=... and B:=...." – Alan Aug 31 '21 at 20:38
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As Ethan Bolker said, there is no general rule for this. My personal recommendation is that, after the definition of $A$ and $B$, you write a little paragraph saying something like:

“After this point and unless specified otherwise, the symbols $A$ and $B$ will stand for the sets defined above.”

That way there is no confusion.

Samuel M. A. Luque
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