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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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    $\begingroup$ @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. $\endgroup$
    – Vidit Nanda
    Commented Jun 5, 2012 at 0:48
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    $\begingroup$ Weil in his 1952 paper credits Borsuk. $\endgroup$
    – Liviu Nicolaescu
    Commented Jun 7, 2012 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
    Commented Jun 7, 2012 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
    Commented Jul 14, 2012 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
    Commented Feb 11, 2015 at 21:19

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