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

$<Y>$ into the non-empty contractible subsets of Y such that

$\varnothing \ne A\subset B\in <Y>$ implies F(A)

$\subset F\left(B\right)$. A set

$Z\subset Y$ is called an F-set if F(A)

$\subset Z$ whenever

$A\in <Z>$. A c-structure (Y,F) is called an l.c. metric space if (Y,d) is a metric space and

$\{$ $y\in Y|$ $d(y,E)<\u03f5\}$ is an F-set whenever

$\u03f5>0$ 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.