Is there a Dowker space of cardinality ${\mathrm{\aleph }}_1$ in ZFC? (open)

A Dowker space is a normal Hausdorff space $X$ such that $X\times [0,1]$ is not normal. By Dowker's theorem (1951) these are exactly the normal spaces that fail to be countably paracompact, so they are the witnesses that normality is not preserved by so much as multiplying with the unit interval.

Question. Does ZFC prove that a Dowker space of cardinality ${\mathrm{\aleph }}_1$ exists?

Status: open. Dowker spaces themselves exist in ZFC, but only three constructions are known: M. E. Rudin's of 1971, of cardinality ${\mathrm{\aleph }}_{\omega }^{{\mathrm{\aleph }}_0}$; Balogh's of 1996, of cardinality the continuum; and the Kojman-Shelah space of 1998, of cardinality ${\mathrm{\aleph }}_{\omega +1}$, obtained from pcf theory as a subspace of Rudin's. Small ones exist under extra axioms: a Dowker space of cardinality ${\mathrm{\aleph }}_1$ can be built under the continuum hypothesis (Juhász, Kunen and Rudin, 1976), from the existence of a Luzin set (Todorcevic), and from the guessing principle $\mathrm{♣}$ (de Caux, 1977; note $\mathrm{♢}$ implies $\mathrm{♣}$, so it suffices too). Whether ZFC alone suffices is not known; this is Conjecture 4 of Rudin's 1990 problem list, and it is still open.

Not proved in this library, and not proved anywhere. The library now develops the required general-topology background, but it does not build a Dowker-space construction or the forcing and independence machinery needed to analyse the ${\mathrm{\aleph }}_1$ question.

What is known, and what would settle it. Settling it means either a ZFC construction of a Dowker space of size ${\mathrm{\aleph }}_1$, or a model of ZFC containing no such space. The consistency of the negative side is what the extra-axiom constructions do not rule out, and it is why the question is genuinely open rather than merely unresolved by the current constructions. Work since has gone on widening the hypotheses that suffice rather than removing them: Rinot, Shalev and Todorcevic derive the relevant guessing principle at ${\mathrm{\aleph }}_1$ from the stick principle, from $\mathrm{♢}(\mathfrak{b})$, and from the existence of a Luzin set. Note the difference in status from the ambient theory: that a ZFC Dowker space exists at all was settled in 1971, and the open part is entirely about how small it can be forced to be.

Why it matters here. The library's separation-axiom material has to record that normality is badly behaved: not hereditary, not productive, and not even stable under multiplication by $[0,1]$. Dowker spaces are the canonical witness for the last of these, and they are also a clean example of a question whose answer is a cardinal rather than a yes or no. Recording it here keeps that material from asserting anything about the smallest such space.