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  "body": "\n## Definition\n\nLet $(X, d_X)$ and $(Y, d_Y)$ be [[metric spaces|063W5JMY85]].\nThe product metric on the [[Cartesian product|0C81YBVTWH]] $X \\times Y$ is the function $d$ with\n$$d\\bigl((x_1, y_1), (x_2, y_2)\\bigr) = \\max\\bigl(d_X(x_1, x_2), \\, d_Y(y_1, y_2)\\bigr)$$\nfor all $x_1, x_2 \\in X$ and $y_1, y_2 \\in Y$.\n\n## Remarks\n\nThe function $d$ is a metric on $X \\times Y$.\nOther common product metrics are $d_X + d_Y$ and $\\sqrt{d_X^2 + d_Y^2}$. Each of them is between $d$ and $2d$, so all of them are [[equivalent|0TWQ5PRKWS]] and give the [[product topology|0Z57EC6YX3]].\nFor $X = Y = \\mathbb{R}$ with the usual metric, $d$ is the [[sup metric|0VTAXQEWQD]] on $\\mathbb{R}^2$.\nThe same formula with a maximum over $n$ factors gives a product metric on a product of $n$ metric spaces.\n\n## Explanations\n\n### L1 human\n\nTwo pairs are close in the product metric when their first parts are close and their second parts are close.\nThe maximum is the worse of the two distances.\n",
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