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The indexed open covering theorem. (English) Zbl 0804.54012

Given an indexed open cover \(\{U_ s : s \in S\}\) of the Tikhonov cube \(I^ S\) (where \(I = [0,1])\), there exists an index \(s \in S\) such that \(U_ s\) contains a connected subset which intersects both \(I^ +_ s = \{x_ s = 1\}\) and \(I_ s^ - = \{x_ s = - 1\}\). The author proves this statement called the ‘indexed open covering theorem’ and derives many important corollaries such as a new proof of the Bohl-Brouwer fixed point theorem.

MSC:

54B10 Product spaces in general topology
54H25 Fixed-point and coincidence theorems (topological aspects)
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