Dowker space

Remarks

By a theorem of Dowker, a normal space $X$ is countably paracompact if and only if the product $X \times [0, 1]$ is normal. So a Dowker space is the same as a normal Hausdorff space $X$ with $X \times [0, 1]$ not normal. Another equivalent condition for a normal space to be countably paracompact: for every decreasing sequence $F_1 \supseteq F_2 \supseteq \cdots$ of closed sets with empty intersection there are open sets $U_n \supseteq F_n$ with empty intersection. An example is the Dowker space constructed by Rudin in ZFC; the check of its properties is long. A non-example is the real line $\R$: it is metrizable, hence paracompact, hence countably paracompact, so clause 3 fails.