(untitled) (Journal of Mathematical Study, 2018)
In this paper, we first show that for every mapping f from a metric space Ω to itself which is continuous off a countable subset of Ω, there exists a nonempty closed separable subspace S ⊂ Ω so that f|_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(Γ) (for any set Γ) 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(Γ) has a fixed point.