Dowker spaces exist in ZFC, and the aleph-one case is open

Dowker's theorem (1951) characterises the normal spaces whose product with the unit interval is again normal: for a normal $X$, the product $X\times [0,1]$ is normal if and only if $X$ is countably paracompact. A Dowker space is a normal space that is not countably paracompact, that is, a witness that the hypothesis cannot be dropped.

(a) Dowker spaces exist in ZFC. M. E. Rudin (1971) gives one, a subspace of a product of ordinal spaces; its cardinality is ${\mathrm{\aleph }}_{\omega }^{{\mathrm{\aleph }}_0}$.

(b) Smaller ZFC examples. Balogh (1996) constructs a Dowker space of cardinality the continuum. Kojman and Shelah (1998) construct one of cardinality ${\mathrm{\aleph }}_{\omega +1}$, using pcf theory.

(c) The ${\mathrm{\aleph }}_1$ case is open. Whether ZFC alone proves the existence of a Dowker space of cardinality ${\mathrm{\aleph }}_1$ is not known. Such spaces are known to exist under extra hypotheses, the standard ones being the continuum hypothesis and the diamond principle.

• Not proved in this library. The library now has a substantial general- topology track, including separation, compactness and paracompactness, but it does not construct a Dowker space or develop the specialised countable-paracompactness and set-theoretic machinery used by the cited constructions.

• What would prove it. All three constructions are ZFC arguments, so no forcing is needed for (a) and (b), but they need ordinal and cardinal arithmetic well past Cardinal (initial ordinal) and cardinality ↗, the ordinal spaces, and in the Kojman-Shelah case Shelah's pcf theory. Clause (c) is an open problem and no track would discharge it.

• Why it matters here. Dowker's theorem is the reason "normal" is not a well-behaved property under products, and a theorem with a hypothesis and no witness is a theorem no reader can calibrate. Recording the ZFC examples is what lets a later separation or product page state Dowker's theorem honestly: the countable paracompactness hypothesis is not removable, and the smallest known ZFC witness is large. Recording (c) is what stops that page from claiming a small witness exists.

• Conditional discipline. (a) and (b) are ZFC theorems, cited and not proved. (c) is a statement about the current state of knowledge, not a mathematical claim, and is recorded so no later page asserts more than is known.