Balogh continuum-sized ZFC Dowker space

Take the space in F1, whose defining map exists by the proved combinatorial construction. F2 and F3 under A1 give its $T_1$, Hausdorff and hereditary-normality properties. F4 gives failure of countable paracompactness. In particular it is normal, and F5 makes it a Dowker space.

The map $\alpha \mapsto (\alpha ,0)$ injects $\kappa$ into $X$. In the reverse direction the underlying set is exactly $\kappa \times \omega$ by F1. The cardinal $\kappa$ is infinite: the binary functions with a single value one at coordinate $n$ give an injection $\omega \to {}^{\omega }2$. Thus F7 gives $\mathrm{\mid }\kappa \times \omega \mathrm{\mid }=\kappa$, under the cardinal identifications allowed by A1. This proves $\mathrm{\mid }X\mathrm{\mid }=2^{{\mathrm{\aleph }}_0}$. Also $X={\bigcup }_{n<\omega }{L}_n$ directly from its underlying set, and F2 gives relative discreteness of each level.

If $X\times [0,1]$ were normal, F6 would apply since $X$ is $T_1$ by step 1.1 and force countable paracompactness, contradicting F4. Hence the product is not normal. Steps 1.1–1.2 give the remaining assertions. Every assumption used was ZFC; no equation identifying the continuum with ${\mathrm{\aleph }}_1$ was needed. QED.