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  "body": "\n## Definition\n\nA *Stone space* is a [[topological space|0CQHFPGJ45]] $X$ such that:\n\n1. $X$ is [[compact|0TVJZYB6SH]].\n2. $X$ is [[Hausdorff|0S1FJ8VEX9]].\n3. $X$ is [[totally disconnected|0RJSBR9GW1]]: its only connected subsets with at least one point are the singletons.\n\n## Remarks\n\nFor a compact Hausdorff space, clause 3 is equivalent to having a basis of [[clopen sets|DQ4E55T3BW]], that is, to being [[zero-dimensional|JJ7D0CYECZ]].\n\nExample: the Cantor space $\\set{0, 1}^{\\N}$ with the product topology.\nIt is compact by Tychonoff's theorem and Hausdorff as a product of Hausdorff spaces.\nTwo different sequences differ at some index $n$, and the set of sequences with a fixed $n$-th term is clopen, so no connected set contains both.\n\nNon-example: the interval $[0, 1]$ is compact and Hausdorff but connected, so clause 3 fails.\nThe space $\\Q$ is Hausdorff and totally disconnected but not compact, so clause 1 fails.\n",
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