Separable Determination of the Fixed Point Property of Convex Sets in Banach Spaces

In this paper, we first show that for every mapping f from a metric space Omega to itself which is continuous off a countable subset of Omega, there exists a nonempty closed separable subspace S subset of Omega so that f vertical bar(S) is again a self mapping on S. Therefore, both the fixed point property and the weak fixed point property of a nonempty closed convex set in a Banach space are separably determined. We then prove that every separable subspace of c(0)(Gamma) (for any set Gamma) is again lying in c(0). Making use of these results, we finally presents a simple proof of the famous result: Every non-expansive self-mapping defined on a nonempty weakly compact convex set of c(0)(Gamma) has a fixed point.