PSEUDOCOMPLETENESS OF COUNTABLE CLOSURES IN SPACES OF CONTINUOUS FUNCTIONS

We answer Question 4.8 of Dow and Tkachuk affirmatively in ZFC: if the closure of every countablesubset of Cp(X) is pseudocomplete, then the Tychonoff space X is a P-space. More precisely, every non-Pspace X admits a closed separable metrizable linear subspace of Cp(X) that is not pseudocomplete. The proofinterpolates sequences at levels tending to zero: finitely supported sequences give continuous functions, whereasthe constant sequence yields the discontinuous characteristic function of the complement of a non-open zeroset. Coordinate evaluation helps make the interpolation image closed, and the Aarts–Lutzer characterizationsupplies the contradiction.