Strength of the effectively Hausdorff-to-effectively discrete subspace principle

Determine the reverse-mathematical strength of the statement that every infinite effectively Hausdorff countable second-countable (CSC) space has an infinite effectively discrete subspace; identify over which subsystems of second-order arithmetic (e.g., RCA0, WKL0, ACA0) this principle is provable.

Background

Earlier sections prove that every infinite Hausdorff CSC space has an infinite discrete subspace (Theorem 5.2) and that every infinite effectively Hausdorff CSC space which is not discrete has an infinite effectively discrete subspace (Theorem 5.3). They also construct a computable effectively Hausdorff CSC space that is discrete but has no infinite computable effectively discrete subspace (Theorem 5.4), and an ω-model where the stronger statement fails (Corollary 5.5).

Despite these results, the precise strength of the universally quantified principle for effectively Hausdorff CSC spaces remains undetermined within the reverse mathematics hierarchy.