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  "body": "\n## Definition\n\nLet $X$ be a set and let $d_1$ and $d_2$ be metrics on $X$.\nThe metrics $d_1$ and $d_2$ are equivalent when they have the same [[open sets|01ZQQDCERP]], that is, when they give the same [[metric topology|0A6YRNZB2R]] on $X$.\nThe metrics are strongly equivalent when there are real numbers $a \u003e 0$ and $b \u003e 0$ with\n$$a \\, d_1(x, y) \\le d_2(x, y) \\le b \\, d_1(x, y) \\quad \\text{for all } x, y \\in X.$$\n\n## Remarks\n\nStrongly equivalent metrics are equivalent: each ball for one metric contains a ball with the same centre for the other metric.\nOn $\\mathbb{R}^n$, the [[taxicab metric|07CJNE6ZAQ]], the [[Euclidean metric|0E1MMCAY5B]] and the [[sup metric|0VTAXQEWQD]] are strongly equivalent.\nOn $\\mathbb{R}$, the metrics $|x - y|$ and $\\min(|x - y|, 1)$ are equivalent but not strongly equivalent, because only the second one is bounded.\nEquivalent metrics have the same convergent sequences and the same continuous maps. Completeness can change: see the Remarks of [[complete metric space|08EWT7J0SC]].\n\n## Explanations\n\n### L1 human\n\nEquivalent metrics can give different numbers for distances, but they agree on which points are near each other.\nFor example, on the plane, the taxicab distance is never more than $\\sqrt{2}$ times the straight distance, and never less than it.\n",
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