Abstract: | 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 $c_0(\Gamma)$ (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. |