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Recently I am writing something up for possible publication. I am struggling with this question:

What is the most appropriate way to cite others' results as a Lemma?

For example, this is a theorem of Arhangelskii's:

Theorem 3.7 Suppose that $X$ is a regular space with a strong rank 1-digonal. Then any bounded subset $Y$ of $X$ is a Moore space.

I want to cite his theorem, however I don't need so much. (Note that if $X$ is a pseudocompact, then it is bounded.) I think if I cite the original theorem, it seems wordy, since I will need to explain not only pseudocompactness but also bounded set. Indeed, I want to write a Lemma as follows:

Lemma [I will note where it is from]: Suppose that $X$ is a regular pseudocompact space with a strong rank 1-diagonal. Then $X$ is a Moore space.

Could I directly do this?

Thanks for your help.

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    $\begingroup$ You're making your life difficult for no reason. Just cite the theorem as Arhanglskii wrote it, and explain in your text why it applies. There's no reason to narrow the scope of the theorem, unless you intend to write a simpler proof that only applies to the special case. $\endgroup$ – dfeuer Jun 7 '13 at 6:55
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    $\begingroup$ Also, Based on your spelling in this post, I strongly recommend that you ask a colleague to copy-edit your article before you submit it for review. $\endgroup$ – dfeuer Jun 7 '13 at 6:57
  • $\begingroup$ @dfeuer: In his theorem, he mentioned another definition of bounded set. If I cite it, I should write its definition. It seems wordy. $\endgroup$ – Paul Jun 7 '13 at 6:59
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    $\begingroup$ I'm not familiar with the specific topic, but your proof will be incomplete if you don't justify the application. Hiding that gap by pretending he proved a theorem he didn't state is not going to help you. Better to be wordy. $\endgroup$ – dfeuer Jun 7 '13 at 7:10
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Either:

  1. the fact that "$X$ is pseudocompact implies $X$ is bounded in the Arhangelskii sense" is well-known, or can be found in the literature, or
  2. it is not.

In the first case you should instead state your Lemma without attribution to Arhangelskii and write

Proof. Since a pseudocompact space is bounded in the sense of Arhangelskii [citation needed], the conclusion is a easy corollary of Theorem 3.7 of [Arhangelskii's paper].

In the second case you absolutely must give the definition of boundedness and give the proof that pseudocompactness implies it.

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  • $\begingroup$ It is nice; +1:) $\endgroup$ – Paul Jun 7 '13 at 9:28
  • $\begingroup$ > Theorem 3.7 of [Arhangelskii's paper] -- Note that many people do not like (and many style guides advise against) using references/citation as nouns. That is, writing "Theorem 3.7 of [Arh98]" is frowned upon, compared to "Theorem 3.7 in Arhangelskii's paper [Arh98]". (E.g. on Wikipedia, the reference numbers [1], [2] etc. are even typeset superscript, and are not considered part of the sentence.) $\endgroup$ – ShreevatsaR Jun 7 '13 at 12:15

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