  We formulate a multi-valued version of the Tietze-Urysohn extension theorem.
Precisely, we prove that any upper semicontinuous multi-valued map with
nonempty closed convex values defined on a closed subset (resp. closed
perfectly normal subset) of a completely normal (resp. of a normal) space $X$
into the unit interval $[0,1]$ can be extended to the whole space $X$. The
extension is upper semicontinuous with nonempty closed convex values. We apply
this result for the extension of real semicontinuous functions, the
characterization of completely normal spaces, the existence of Gale-Mas-Colell
and Shafer-Sonnenschein type fixed point theorems and the existence of
equilibrium for qualitative games.
