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  "body": "\n## Statement\n\nLet $(X, d)$ be a [[metric space|063W5JMY85]], with the [[metric topology|0A6YRNZB2R]].\nThen $X$ is a [[Hausdorff space|0MXB4CKHFW]].\n\n## Remarks\n\nProof: for $x \\neq y$, let $r = d(x, y) / 2 \u003e 0$. The [[open balls|0DC3MPR7C9]] $B(x, r)$ and $B(y, r)$ are open. They are disjoint: a point $z$ in both would give $d(x, y) \\le d(x, z) + d(z, y) \u003c 2r$, by the [[triangle inequality|01M6CW9VM6]].\nThus each [[metrizable space|09KDHB82SG]] is Hausdorff. A non-Hausdorff space, such as an infinite set with the cofinite topology, is not metrizable.\nFor a pseudometric, the statement fails: points at distance $0$ cannot be separated.\n\n## Explanations\n\n### L1 human\n\nTwo distinct points of a metric space are at a positive distance. Balls of half that radius about them do not meet.\n",
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