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  "body": "\n## Counterexample\n\nThe Sorgenfrey plane is the product $\\R_\\ell \\times \\R_\\ell$ of two copies of the [[Sorgenfrey line|Y4T3GVK8GV]] with the [[product topology|0GBG8B49AA]].\nEach factor is normal, but the Sorgenfrey plane is not [[normal|0NSR9V8TED]].\n\n## Remarks\n\nThe basic open sets are the half-open rectangles $[a, b) \\times [c, d)$.\nThe anti-diagonal $D = \\set{(x, -x) : x \\in \\R}$ is [[closed|0ZCK8DQYN1]], since it is closed in the coarser Euclidean topology.\nIt is discrete: the rectangle $[x, x + 1) \\times [-x, -x + 1)$ meets $D$ only in $(x, -x)$.\nThe plane is [[separable|0Z2P41SVDP]], because $\\Q \\times \\Q$ is dense.\nBy Jones' lemma, a separable normal space has no closed discrete subset $D$ with $2^{\\abs{D}} \u003e 2^{\\aleph_0}$.\nHere $\\abs{D} = 2^{\\aleph_0}$, so the plane is not normal.\nConcretely, the points of $D$ with rational first coordinate and those with irrational first coordinate form two disjoint closed sets that no disjoint open sets separate.\nThe plane also shows that a product of two [[Lindelöf|SW0SN1NEJE]] spaces need not be Lindelöf.\n\n## Explanations\n\n### L1 human\n\nThe Sorgenfrey line is a well-behaved space.\nMultiplying it by itself gives a plane in which two disjoint closed sets cannot be pulled apart by open sets.\nSo normality is a property that does not survive products.\nThe trouble sits on the line $y = -x$, where the half-open rectangles touch the line in only one point each.\n",
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