Optimality of neighborhood-condition coding for partial continuous functions

Prove that the neighborhood-condition coding of partial continuous functions between complete separable metric spaces is optimal, in the sense that it provides an optimal coding of continuous functions defined on a $G_{\delta}$ subset.

Background

The paper discusses whether the coding of higher-order mathematical objects in reverse mathematics preserves the intended logical strength of the corresponding ordinary mathematical theorems. In this context, it cites a problem concerning the standard neighborhood-condition representation of partial continuous functions between complete separable metric spaces.

The problem asks for a mathematical justification of the optimality of this representation, specifically because it amounts to coding continuous functions on a GδG_{\delta} subset. The paper notes that its results on alternative representations of open sets challenge the idea that the usual second-order coding is optimal, but does not resolve the cited optimality problem.

References

PROBLEM. Continuation of the previous problem: Show that [the] neighborhood condition coding of partial continuous functions between complete separable metric spaces is “optimal”. (It amounts to a coding of continuous functions on a $G_{\delta}$.)

— Coding is non-robust  (2609.18371 - Sanders, 16 Sep 2026) in Section 6.2, subsection “The coding practice and issue of RM” (quoting Friedman and Simpson)