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  "body": "\n## Counterexample\n\nLet $Y = \\R \\times \\set{0, 1}$, the disjoint union of two copies of the real line.\nIdentify $(x, 0)$ with $(x, 1)$ for every $x \\ne 0$, and give the set $X$ of classes the quotient topology.\nThe two classes $0_a$ of $(0, 0)$ and $0_b$ of $(0, 1)$ are the two origins.\nThe space $X$ is locally homeomorphic to $\\R$, [[second-countable|03PPNCM9GG]] and $T_1$, but it is not [[Hausdorff|0S1FJ8VEX9]].\n\n## Remarks\n\nRemoving $0_b$ leaves an open set homeomorphic to $\\R$, and the same holds for $0_a$; these two open sets cover $X$.\nEvery [[neighbourhood|0ZCY0Z80B6]] of $0_a$ contains the image of $(-\\varepsilon, 0) \\cup (0, \\varepsilon)$ for some $\\varepsilon \u003e 0$, and so does every neighbourhood of $0_b$.\nSo any neighbourhood of $0_a$ meets any neighbourhood of $0_b$, and the two origins cannot be separated.\nFor the same reason the sequence of classes of $1/n$ [[converges|0W4CN9E2W5]] to both $0_a$ and $0_b$, so limits of sequences are not unique.\nThis is why the usual definition of a [[topological manifold|QQQ9B1DK4P]] requires the Hausdorff condition: without it, this space would count as a $1$-manifold.\n\n## Explanations\n\n### L1 human\n\nTake the real line and double its zero point, keeping everything else the same.\nNear each point, the new space still looks like a piece of the line.\nBut the two zeros are glued so tightly to the same nearby points that no pair of open sets can keep them apart.\nA sequence that goes to zero now goes to both zeros at once.\n",
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