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
. A set
is called an F-set if F(A)
. A c-structure (Y,F) is called an l.c. metric space if (Y,d) is a metric space and
is an F-set whenever
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.