  The famous Banach Contraction Principle holds in complete metric spaces, but
completeness is not a necessary condition -- there are incomplete metric spaces
on which every contraction has a fixed point. The aim of this paper is to
present various circumstances in which fixed point results imply completeness.
For metric spaces this is the case of Ekeland variational principle and of its
equivalent - Caristi fixed point theorem. Other fixed point results having this
property will be also presented in metric spaces, in quasi-metric spaces and in
partial metric spaces. A discussion on topology and order and on fixed points
in ordered structures and their completeness properties is included as well.
