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  "body": "\n## Example\n\nLet $M_0 = [0, 1]^3$.\nDivide each cube of side $3^{-k}$ in $M_k$ into $27$ equal cubes of side $3^{-k-1}$.\nRemove the interiors of the central cube and of the $6$ cubes that share a face with it.\nThe union of the other $20$ cubes is $M_{k+1}$.\nThe *Menger sponge* is $M = \\bigcap_{k \\ge 0} M_k$.\n\nThe set $M$ is a [[continuum|NJQ196M1WR]].\nEach $M_k$ is a finite union of closed cubes, so it is [[compact|0TVJZYB6SH]].\nIt is [[connected|0V032VWTHD]]: the $20$ kept cubes in each divided cube are the $12$ edge cubes and $8$ corner cubes, which form a connected frame.\nA decreasing intersection of nonempty compact connected sets is nonempty, compact and connected.\n\n## Remarks\n\nThe set $M_k$ has volume $(20/27)^k$, so $M$ has volume $0$.\nIts Hausdorff dimension is $\\log 20 / \\log 3 \\approx 2.727$.\nIts [[Lebesgue covering dimension|A3W7RGXXK9]] is $1$, so topologically it is a curve.\nEach face of the cube meets $M$ in a [[Sierpiński carpet|PCM3H6J3EE]].\nMenger showed that every compact metric space of dimension at most $1$ embeds in $M$, so $M$ is a universal curve.\n\n## Explanations\n\n### L1 human\n\nTake a cube, cut it into a three by three by three grid, and remove the middle cube and the six cubes in the middle of the faces.\nRepeat with each of the twenty small cubes left, and keep going forever.\nThe result is a sponge with tunnels at every scale.\nIt has no volume and it is made of curves only, yet every curve one can imagine fits inside it.\n",
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