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  "body": "\n## Definition\n\nLet $(X, d)$ be a [[metric space|0VKD1TP7F3]], and for $s \\ge 0$ let $\\mathcal{H}^s$ be the $s$-dimensional [[Hausdorff measure|Q1TGZK8RN6]] on $X$.\nThe *Hausdorff dimension* of a set $A \\subseteq X$ is\n$$\\dim_H A = \\inf \\set{ s \\ge 0 : \\mathcal{H}^s(A) = 0 },$$\nwith the convention $\\inf \\emptyset = \\infty$.\n\n## Remarks\n\nIf $\\mathcal{H}^s(A) \u003c \\infty$ and $t \u003e s$, then $\\mathcal{H}^t(A) = 0$.\nSo $\\mathcal{H}^s(A) = \\infty$ for $s \u003c \\dim_H A$ and $\\mathcal{H}^s(A) = 0$ for $s \u003e \\dim_H A$.\nAt $s = \\dim_H A$ the value can be $0$, positive and finite, or $\\infty$.\nEquivalently, $\\dim_H A = \\sup \\set{s \\ge 0 : \\mathcal{H}^s(A) = \\infty}$, with $\\sup \\emptyset = 0$.\n\nExample: the middle-thirds [[Cantor set|0BXNDZFQME]] $C$ has $\\dim_H C = \\log 2 / \\log 3$.\nAt the $k$-th step, $C$ is covered by $2^k$ intervals of length $3^{-k}$, so $\\mathcal{H}^s_{3^{-k}}(C) \\le 2^k 3^{-ks}$, which equals $1$ for $s = \\log 2 / \\log 3$.\nThis gives $\\dim_H C \\le \\log 2 / \\log 3$; the matching lower bound needs a separate argument.\n\nA dense set can have dimension $0$: $\\Q$ is dense in $\\R$, yet $\\dim_H \\Q = 0$.\nEach point has $\\mathcal{H}^s$ measure $0$ for $s \u003e 0$, and $\\mathcal{H}^s$ is countably subadditive.\n\nFor subsets of $\\R^n$, one has $\\dim_H A \\le n$, and every nonempty open set has dimension $n$.\n\n## Explanations\n\n### L1 human\n\nThe Hausdorff dimension is the exponent $s$ at which the $s$-dimensional size of a set drops from infinite to zero.\nIt measures how a set fills space at small scales.\n",
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