On Kuhns Strong Cubical Lemma

Cover On Kuhns Strong Cubical Lemma
On Kuhns Strong Cubical Lemma
Robert Michael Freund
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, M} . Let v be a limit point of a subsequence of an infinite sequence of such x for asequence of triangulations C of S whose mesh goes to zero. Then since each set Cr. Is closed k = 1, ... , m. - 1, k j j = 1, ... , n, and C, " is closed, v € n C~ n C. , proving the theorem. H j=l, . . . , n k=l, . . . M. J-1 Paralleling the covering theorem of Knaster et. Al. , we have: The Simplotope Covering Theorem implies Brouwer's theorem : To see this, let f : S ■+ S be a given continuous function. Defi...ne the following sets: -4. 6- cl :(veS| f*(v) Svjj'. K-l n. , -1, k c£ ■■ {v e S I f fc (v) ; vj. , fj ± (v) :! v m±, 1 ■ 1 ] ■■!}, j-2 n, k=l, ... , m. — 1, 3 , n+l C^ = {v e S | f^(v) j £ n+1 m i and m v m .

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