[en] This poster studies a notion arising from the theory of interpolation spaces extended
to metric spaces. To compensate for the lack of algebraic structure in this setting, the
constructions introduced in [1, 3] share a common underlying principle, leading to a natural way of extending a metric space A within a metric space B.
This extension procedure has never been investigated in a systematic framework. As a
consequence, some fundamental questions remain unanswered. In particular, it is referred
to as a “completion”, although it can yield a non-complete space. Therefore, existing
approaches provide only a partial understanding of the construction. In this work, we
develop a general theory of this procedure which we call the Cauchy extension. In
particular, we will present an example of a metric space with a non-complete Cauchy
extension. Furthermore, in analogy with the Cantor-Bendixson theorem [2], we will provide a genuine completion procedure by showing that iterating the Cauchy extension along ordinals always stabilizes at a complete metric space.
