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  "body": "\n## Definition\n\nLet $X$ be a [[topological space|0CQHFPGJ45]], and write $\\partial V$ for the [[boundary|05D5EGMFYV]] of a set $V \\subseteq X$.\nThe *small inductive dimension* $\\operatorname{ind} X$ is defined by induction on $n \\ge -1$:\n\n1. $\\operatorname{ind} X = -1$ if and only if $X = \\emptyset$.\n2. For $n \\ge 0$, $\\operatorname{ind} X \\le n$ if for every point $x \\in X$ and every [[open set|02BH3MCBQB]] $U$ with $x \\in U$ there is an open set $V$ with $x \\in V \\subseteq U$ and $\\operatorname{ind} \\partial V \\le n - 1$.\n3. $\\operatorname{ind} X = n$ if $\\operatorname{ind} X \\le n$ and not $\\operatorname{ind} X \\le n - 1$; and $\\operatorname{ind} X = \\infty$ if $\\operatorname{ind} X \\le n$ holds for no $n$.\n\nThe *large inductive dimension* $\\operatorname{Ind} X$ is defined in the same way, with a new clause 2.\nFor $n \\ge 0$, $\\operatorname{Ind} X \\le n$ if for every [[closed set|0ZCK8DQYN1]] $A$ and every open set $U \\supseteq A$ there is an open set $V$ with $A \\subseteq V \\subseteq U$ and $\\operatorname{Ind} \\partial V \\le n - 1$.\nHere the boundary $\\partial V$ carries the subspace topology.\n\n## Remarks\n\nThe rational numbers $\\Q$ have $\\operatorname{ind} \\Q = 0$.\nGiven $x \\in U$, choose irrational numbers $a \u003c x \u003c b$ with $(a, b) \\cap \\Q \\subseteq U$; then $V = (a, b) \\cap \\Q$ is open and closed in $\\Q$, so $\\partial V = \\emptyset$.\nThe real line has $\\operatorname{ind} \\R = 1$.\nSmall open intervals have a two-point boundary, which has dimension $0$, so $\\operatorname{ind} \\R \\le 1$.\nBut $\\operatorname{ind} \\R \\le 0$ fails: a nonempty open set $V \\neq \\R$ has nonempty boundary, because $\\R$ is connected.\nFor separable metric spaces, $\\operatorname{ind} X = \\operatorname{Ind} X = \\dim X$, where $\\dim$ is the [[Lebesgue covering dimension|A3W7RGXXK9]].\n\n## Explanations\n\n### L1 human\n\nA space has inductive dimension at most $n$ when small pieces of it can be cut out by walls of dimension at most $n - 1$.\n",
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