Characterize compact metric spaces reconstructible by a finite approximative sequence (fas)

Characterize the class of compact metric spaces X that admit a finite approximative sequence (fas)—that is, an inverse sequence of finite T0 spaces U_{2ε_n}(A_n) built from finite ε_n-approximations A_n of X with the bonding maps p_{n,n+1} defined by nearest-point selections—whose inverse limit is homeomorphic to X.

Background

The paper adapts a homotopy reconstruction theorem to construct inverse sequences of finite T0 spaces (finite approximative sequences, or fas) from ε-approximations of a compact metric space X. For general compact metric spaces, previous work shows that the inverse limit of such a sequence contains a copy of X as a strong deformation retract, yielding a homotopy (but not necessarily homeomorphic) reconstruction.

The main new results establish that two broad classes—countable compact metric spaces and compact ultrametric spaces—are fully reconstructible: their topology is recovered exactly as an inverse limit of finite T0 spaces arising from the fas construction. The concluding question asks for a precise characterization of all compact metric spaces that enjoy this stronger, topological reconstruction property, beyond the classes proven in the paper.