I am sorry if this question is not for mathoverflow. I asked the same question in tex stackexchange and got an answer that this question is out of topic. If it's the same here, please let me know.

There are two setbuilder notations, the vertical bar and the colon. In some cases, it is better to make a choice. For example, compare $$\{f \mid f\colon M \rightarrow N \text{ is continuous}\}$$ and $$\{f : f\colon M \rightarrow N \text{ is continuous}\}.$$ Also compare $$\{x : \lvert x \rvert = 1\}$$ and $$\{x \mid \lvert x \rvert = 1\}.$$

Can I use both the vertical bar and the colon in one article to make it more readable? Or should I choose one notation to be consistent?

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    $\begingroup$ I vote for the colon, as it's not as overloaded as the vertical bar. I also think you should be consistent, whichever you choose. $\endgroup$ – Eric Tressler Apr 23 '13 at 13:35
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    $\begingroup$ Shouldn't this question rather be community wiki? $\endgroup$ – Stefan Kohl Apr 23 '13 at 13:48
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    $\begingroup$ People should use the vertical bar to avoid frowny faces as well. For example, you should write $\{(x,y)|x^{2}+y^{2}=1\}$ instead of $\{(x,y):x^{2}+y^{2}=1\}$. Editors do not like it when papers are filled with negative emoticons. $\endgroup$ – Joseph Van Name Apr 23 '13 at 15:30
  • $\begingroup$ I have sometimes seen semicolon used instead of colon or vertical bar, but I very much prefer the bar anyway, even if it gets ugly. $\endgroup$ – Ketil Tveiten Apr 23 '13 at 16:56
  • $\begingroup$ I like using "such that" (i.e., \text{such that}) whenever there's room for it. And with semicolons, it might seem like the paper is sadly winking ); $\endgroup$ – Marty Apr 23 '13 at 18:57

As the author, such things are generally up to you. It would not be the first time an author chose clarity over consistency of notation. I think many readers appreciate such tradeoffs, although I am sure there is someone who doesn't. If you are going to use inconsistent notation, it is polite to make a footnote or something about why you have done so, but this case is probably transparent enough to make that unnecessary.

If using both really bothers you, you could use \big to do things like $\{ x\ \big\vert\ \lvert x\rvert = 1\}$, which is slightly more readable than when all the symbols are the same size.

  • $\begingroup$ Boy, this is the best answer I have seen in MO. I have learned a lot from other answers, but I was surprised to read more and more detail in only 7 lines; +2! (I wish). I would add that the two notations look natural enough that the reader will probably not notice if they are both used. $\endgroup$ – Rodrigo A. Pérez Apr 23 '13 at 13:46

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