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  "body": "\n## Counterexample\n\nLet $\\omega_1$ be the first uncountable [[ordinal|0YSCSX5YZC]].\nThe closed long ray is the set $L = \\omega_1 \\times [0, 1)$ with the [[lexicographic order|0B68KNPGCG]] and the order topology.\nThe long line is obtained by gluing two copies of $L$ along their least point $(0, 0)$, one of them with the reversed order.\nThe closed long ray $L$ is Hausdorff, [[connected|0V032VWTHD]] and [[sequentially compact|0FRMJH3CY7]], but it is not [[compact|0TVJZYB6SH]] and not [[metrizable|091GJ3FYD7]].\n\n## Remarks\n\nFor each countable ordinal $\\beta \\ge 1$, the initial segment of $L$ below $(\\beta, 0)$ is a countable stack of half-open intervals, and it is homeomorphic to $[0, 1)$.\nSo every point of $L$ other than $(0, 0)$ has a neighbourhood homeomorphic to an open interval of $\\R$.\nA sequence $(\\alpha_n, t_n)$ in $L$ has a countable supremum $\\gamma$ of the $\\alpha_n$, because a countable union of countable sets is countable.\nSo the sequence lies in the set of points up to $(\\gamma + 1, 0)$, which is homeomorphic to $[0, 1]$, and it has a convergent subsequence.\nThe open sets $\\set{p \\in L : p \u003c (\\alpha, 0)}$ for $\\alpha \u003c \\omega_1$ cover $L$, but no finitely many of them do, because $\\omega_1$ has no largest element.\nA sequentially compact metrizable space is compact, so $L$ is not metrizable.\nFor the same reason the long line is a connected Hausdorff space, locally homeomorphic to $\\R$, that is not second-countable and not metrizable.\nSo being Hausdorff and locally Euclidean does not make a space a [[topological manifold|QQQ9B1DK4P]] in the usual sense, which also asks for second countability.\n\n## Explanations\n\n### L1 human\n\nThe long line is made by laying uncountably many copies of the interval $[0, 1)$ end to end.\nNear each of its points it looks just like the real line.\nBut it is so long that no sequence can climb to its end.\nIt shows that \"looks like a line near every point\" does not give a space that can be measured by a distance.\n",
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