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  "body": "\n## Statement\n\nLet $X$ be a [[topological space|0CNQXVBHCX]] and let $x \\in X$.\nThe connected component of $x$ is the [[union|05GK32H72H]] $C(x)$ of all [[connected subsets|0R37DFJKFW]] of $X$ that contain $x$.\nThen $C(x)$ is connected and [[closed|0G5AV9Q4RR]] in $X$.\n\n## Remarks\n\nProof: a union of connected sets with a common point is connected, so $C(x)$ is the largest connected set that contains $x$. The [[closure|0DRKT1QJR7]] of $C(x)$ is also connected, so it equals $C(x)$.\nThe components of $X$ form a partition of $X$.\nComponents need not be open. In $\\mathbb{Q}$, the components are the single points, which are not open.\nIf $X$ has only finitely many components, then each of them is also open.\n\n## Explanations\n\n### L1 human\n\nThe component of a point is the largest piece of the space that contains the point and is in one piece.\nTaking the closure cannot make the piece larger, so each component is closed.\n",
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