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Recently, I've read two papers which have cited the Nerve Theorem, one crediting Borsuk with the result and another Leray. Here is the question:

Who was the first to prove the Nerve Theorem?

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  • $\begingroup$ In what sense do you mean they proved the nerve theorem? Borsuk or Leray certainly didn't think of it in the terms that we use. Could you give us references as to where Borsuk/Leray are supposed to have proved it? $\endgroup$ – David Roberts Jun 4 '12 at 23:41
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    $\begingroup$ Weil in his 1952 paper credits Borsuk. $\endgroup$ – Liviu Nicolaescu Jun 7 '12 at 8:21
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    $\begingroup$ Following up on JeffE's comment: Ken Brown, in his Cohomology of Groups book (Chapter VII.4), says that the homology version of the nerve lemma "seems to be essentially due to Leray". (Presumably the 1945 paper.) In the exercises he discusses the homotopy version, which he attributes to Weil, 1952. $\endgroup$ – Russ Woodroofe Jun 7 '12 at 13:46
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    $\begingroup$ As far as I know, nerves were introduced by Paul Alexandroff in 1928 --- once the notion is introduced I do not see much to prove... $\endgroup$ – Anton Petrunin Jul 14 '12 at 23:15
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    $\begingroup$ McCord's paper is especially clear on this subject, which surveys Weil's work on the nerve theorem that results in a homotopy equivalence, not just same homology. ams.org/journals/proc/1967-018-04/S0002-9939-1967-0216499-0/… $\endgroup$ – Justin Curry Feb 11 '15 at 21:19

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