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Contractibility and generalized convexity. (English) Zbl 0733.54011
The author defines various generalizations of convexity which are strong enough to give rise to selection theorems for set-valued mappings. He then applies these selection theorems to obtain fixed point results for set-valued mappings. A c-structure on the topological space Y is given by a mapping F from the non-empty finite subsets $$ into the non-empty contractible subsets of Y such that $\varnothing \ne A\subset B\in $ implies F(A)$\subset F\left(B\right)$. A set $Z\subset Y$ is called an F-set if F(A)$\subset Z$ whenever $A\in $. A c-structure (Y,F) is called an l.c. metric space if (Y,d) is a metric space and $\left\{$ $y\in Y|$ $d\left(y,E\right)<ϵ\right\}$ is an F-set whenever $ϵ>0$ and E is an F-set and if open balls are F-sets. Michael’s theorem, in this context, reads as follows: Let X be a paracompact space, (Y,F) an l.c. complete metric space, and let T be a lower semicontinuous mapping from X into the non- empty closed F-sets of Y. Then there is a continuous selection for T. The author then derives fixed point theorems and provides a wealth of examples.

##### MSC:
 54C60 Set-valued maps (general topology) 54H25 Fixed-point and coincidence theorems in topological spaces 47H10 Fixed point theorems for nonlinear operators on topological linear spaces 54C65 Continuous selections
minimax theorem