Constructive validity of Tietze extension with closed sets defined as complements of constructive open sets

Determine whether the Tietze Extension Theorem holds in constructive mathematics when closed sets are defined as complements of constructive open sets. Specifically, ascertain whether for every normal topological space X and every continuous constructive function f defined on a closed subset A ⊆ X (where “closed” means A = X \ U for some constructive open set U, and a set U is constructive open if for each point x ∈ U there exists an algorithm that outputs an open ball containing x and contained in U), there always exists a continuous constructive extension F of f to X.

Background

The paper proves that the classical Tietze Extension Theorem fails in a constructive setting when closed sets are interpreted as sequentially closed. Using a discrete metric space on the natural numbers and an unextendible computable {0,1}-valued function, the authors construct sequentially closed sets A and B for which no continuous computable extension to the whole space exists, thereby contradicting the classical extension principle.

They also observe that under this sequentially closed interpretation, Urysohn's lemma fails constructively. Motivated by the sensitivity of extension theorems to how closed sets are defined in constructive topology, the authors pose whether adopting a different constructive notion of closed sets—namely, complements of constructive open sets—could restore the validity of Tietze's theorem.

The paper further clarifies that a constructive open set is one for which, at every point, there exists a program that produces an open ball contained in the set and containing that point, and contrasts this with Lacombe open sets. The open question asks if redefining closed sets in this way changes the constructive status of Tietze extension.