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  "body": "\n## Definition\n\nLet $X$ be a set.\nA metric on $X$ is a [[function|0XHDTE5YMZ]] $d \\colon X \\times X \\to \\mathbb{R}$ with these properties, for all $x, y, z \\in X$:\n\n1. $d(x, x) = 0$.\n2. If $x \\neq y$, then $d(x, y) \u003e 0$.\n3. $d(x, y) = d(y, x)$.\n4. $d(x, z) \\le d(x, y) + d(y, z)$.\n\nA metric space is a pair $(X, d)$, where $X$ is a set and $d$ is a metric on $X$.\nThe number $d(x, y)$ is the distance between $x$ and $y$.\n\n## Remarks\n\nProperty 4 is the [[triangle inequality for a metric|01M6CW9VM6]].\nThe properties give $d(x, y) \\ge 0$ for all $x, y$: take $z = x$ in property 4 and use property 3.\nThe set $\\mathbb{R}$ with $d(x, y) = |x - y|$ is a metric space: this is the [[usual metric on the real line|0KXJTZSW9B]].\nIf you remove property 2, you get a [[pseudometric space|0H3W614GEC]].\n\n## Explanations\n\n### L1 human\n\nA metric is a rule that gives a distance to each pair of points.\nThe four properties are the facts that you expect from distances on a map.\nFor example, the road distance between towns is not always a metric, because one-way roads can break symmetry.\n",
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