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  "body": "\n## Summary\n\nThe Lebesgue covering dimension of a topological space is the least $n$ with the following property.\nEvery finite open cover can be refined to an open cover in which no point lies in more than $n + 1$ sets.\nIt is a topological invariant, and it gives $n$ for the Euclidean space $\\R^n$.\nIt is one of the main notions of dimension theory, together with the [[inductive dimensions|MKS9H497AM]].\n\n## Explanations\n\n### L1 human\n\nCover a line segment by short overlapping open intervals; one can arrange that each point lies in at most two of them, but not always in only one.\nCover a square by small open tiles; one can arrange that each point lies in at most three of them, like bricks in a wall, but not in fewer.\nThe covering dimension turns this count into a definition: a space has dimension $n$ when the best one can do is $n + 1$.\nSo a segment has dimension $1$ and a square has dimension $2$.\n",
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