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  "body": "\n## Counterexample\n\nLet $X = \\N \\times \\N$, where $\\N = \\set{0, 1, 2, \\dots}$, and call $\\set{m} \\times \\N$ the $m$-th column.\nDeclare a set $U \\subseteq X$ open if either $(0, 0) \\notin U$, or $(0, 0) \\in U$ and for all but finitely many $m$ the set $\\set{m} \\times \\N \\setminus U$ is finite.\nThis is the Arens–Fort space.\nIt is a countable [[Hausdorff|0S1FJ8VEX9]] and [[normal|0NSR9V8TED]] space.\nThe point $(0, 0)$ lies in the [[closure|0QEAJB91GZ]] of $X \\setminus \\set{(0, 0)}$.\nBut no sequence of points different from $(0, 0)$ [[converges|0W4CN9E2W5]] to $(0, 0)$.\nSo it is not a [[sequential space|WMB2MAEDX4]] and not [[first-countable|0CP5NDD21C]].\n\n## Remarks\n\nEvery point other than $(0, 0)$ is isolated, and every open set containing $(0, 0)$ meets $X \\setminus \\set{(0, 0)}$, so $(0, 0)$ is in the closure.\nLet $(p_k)$ be a sequence in $X \\setminus \\set{(0, 0)}$ with infinitely many distinct terms, and let $S$ be the set of its terms.\nSuppose some column contains infinitely many points of $S$, and remove these points from $X$.\nThe rest of $X$ misses infinitely many points of only one column, so it is an open set containing $(0, 0)$.\nOtherwise every column meets $S$ in a finite set, so $X \\setminus S$ is an open set containing $(0, 0)$.\nIn both cases an open set around $(0, 0)$ misses infinitely many terms, so the sequence does not converge to $(0, 0)$.\nA sequence with only finitely many distinct terms, all different from $(0, 0)$, stays outside the open set $X$ minus these terms.\nThus $X \\setminus \\set{(0, 0)}$ is sequentially closed but not closed.\nAn open set containing $(0, 0)$ is also closed, since its complement misses $(0, 0)$, and each $\\set{x}$ with $x \\ne (0, 0)$ is open and closed, so the space is regular.\nA countable regular space is Lindelöf, and a regular Lindelöf space is normal; this gives normality.\n\n## Explanations\n\n### L1 human\n\nPicture an infinite grid of points with one special corner point.\nAll grid points other than the corner are open on their own.\nTo be near the corner, a set must contain almost all of almost every column.\nNo sequence of grid points can be near the corner in this sense, yet the corner still touches the rest of the grid.\nSo sequences fail to detect closure in this space.\n",
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