## 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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