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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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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? –  David Roberts Jun 4 '12 at 23:41
    
I think also that there may be more than one nerve theorem. All the nerve theorems says something like if you have a sufficiently nice covering, then the nerve behaves like the space, but there are different variants of this. –  Benjamin Steinberg Jun 5 '12 at 0:36
    
@David Here are the citations from those papers: K. Borsuk, On the imbedding of systems of compacta in simplicial complexes , Fund. Math 35, (1948) 217-234 J. Leray. Sur la forme des espaces topologiques et sur les points fixes des représentations. J. Math. Pures Appl. 24:95–167, 1945 But it is not clear that these are the first instances of such a result. @Benjamin: I refer to the version which holds for contractible nerves in a paracompact space. –  Vidit Nanda Jun 5 '12 at 0:48
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Weil in his 1952 paper credits Borsuk. –  Liviu Nicolaescu Jun 7 '12 at 8:21
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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. –  Russ Woodroofe Jun 7 '12 at 13:46
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