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  "body": "\n## Definition\n\nLet $X$ be a topological space.\n$X$ is *countably paracompact* if every countable open cover of $X$ has an open refinement that is a [[locally finite family|DZ297MMGV8]].\n$X$ is a *Dowker space* if:\n\n1. $X$ is [[Hausdorff|0S1FJ8VEX9]],\n2. $X$ is [[normal|0NSR9V8TED]], and\n3. $X$ is not countably paracompact.\n\n## Remarks\n\nBy a theorem of Dowker, a normal space $X$ is countably paracompact if and only if the product $X \\times [0, 1]$ is normal.\nSo a Dowker space is the same as a normal Hausdorff space $X$ with $X \\times [0, 1]$ not normal.\nAnother 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.\nAn example is the Dowker space constructed by Rudin in ZFC; the check of its properties is long.\nA non-example is the real line $\\R$: it is metrizable, hence [[paracompact|8R29G6NG9Q]], hence countably paracompact, so clause 3 fails.\n",
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