On the closedness of sets with the fixed point property for contractions - Mira-Cristiana Anisiu; Valeriu Anisiu (Journal of Numerical Analysis and Approximation Theory, 1997)
It is proved by examples that there exist (connected) non-closed sets with the fixed point property for contractions in complete metric spaces. In a Banach space, a convex set with nonvoid interior having the fixed point property for contractions is necessarily closed.