Generic continuity of magnitude along arbitrary sequences (Formulation B)

Determine whether the magnitude mapping X ↦ |X| on the Gromov–Hausdorff space FMet of isometry classes of finite metric spaces is generically continuous in the sense of Formulation B for arbitrary convergent sequences. Specifically, ascertain whether for a generic convergent sequence (X_k) in FMet we have lim_{k→∞} |X_k| = |lim_{k→∞} X_k|, i.e., whether a version of Theorem 4.1 (magnitude behaves continuously for generic limits along lines) extends to a space of all sequences in FMet.

Background

The paper introduces two notions of ‘generic continuity’ for the magnitude invariant of finite metric spaces. Formulation A (continuity on a dense open subset) fails dramatically: the authors prove magnitude is nowhere continuous on FMet. Formulation B reframes generic continuity sequentially, requiring that magnitude preserve limits for generic convergent sequences.

While the authors establish a positive result for a restricted class of paths—lines in Gromov–Hausdorff space—they explicitly note that their methods do not yet address arbitrary sequences. The open question asks whether this sequential generic continuity extends beyond lines to all sequences in FMet, thereby yielding a more comprehensive stability theorem for magnitude.