Completeness of ultrabornological subspaces in complete LB-spaces

Determine whether, for every closed subspace Y of a complete LB-space X, the ultrabornological subspace Y^\flat is complete.

Background

The \flat-topology (ultrabornological topology) on subspaces plays a central role in kernels and images in LB categories. If Y\flat were always complete for closed Y in a complete LB-space, it would strengthen the stability properties of exact structures and functorial constructions in the paper’s homological approach.

References

  1. It is unknown if for a closed subspace $Y\subseteq X$ of a complete LB-space the space $Y{\flat}$ is always complete.
— A homological approach to (Grothendieck's) completeness problem for regular LB-spaces  (2512.13161 - Wegner, 15 Dec 2025) in Section “Complete LB-spaces” (SEC-COM), opening paragraph