Published October 19, 2026 | Version v1
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PSEUDOCOMPLETENESS OF COUNTABLE CLOSURES IN SPACES OF CONTINUOUS FUNCTIONS

Description

We answer Question 4.8 of Dow and Tkachuk affirmatively in ZFC: if the closure of every countable
subset 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 proof
interpolates sequences at levels tending to zero: finitely supported sequences give continuous functions, whereas
the constant sequence yields the discontinuous characteristic function of the complement of a non-open zero
set. Coordinate evaluation helps make the interpolation image closed, and the Aarts–Lutzer characterization
supplies the contradiction.

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References

  • J. M. Aarts and D. J. Lutzer, Pseudo-completeness and the product of Baire spaces, Pacific J. Math. 48 (1973), 1–10
  • Alan Dow and Vladimir V. Tkachuk, Reflecting topological properties in closed separable subspaces, Eur. J. Math. 11 (2025), Art. No. 5.
  • Ryszard Engelking, General Topology, Sigma Series in Pure Mathematics, vol. 6, Heldermann Verlag, 1989
  • J. C. Oxtoby, Cartesian products of Baire spaces, Fund. Math. 49 (1961), 157–166.