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  "body": "\n## Definition\n\nLet $X$ be a [[topological space|0CNQXVBHCX]] and let $S \\subseteq X$.\nThe set $S$ is connected when $S$, with the [[subspace topology|0D5YHQGDJY]], is a [[connected space|0D939CMME7]].\n\n## Remarks\n\nEquivalently, there are no open sets $U, V$ of $X$ with $S \\subseteq U \\cup V$, $S \\cap U \\cap V = \\varnothing$, $S \\cap U \\neq \\varnothing$ and $S \\cap V \\neq \\varnothing$.\nThe [[connected subsets of the real line are the intervals|0B9ZQFMQNV]].\nThe closure of a connected set is connected. A union of connected sets with a common point is connected.\n\n## Explanations\n\n### L1 human\n\nA subset is connected when it is in one piece, as a space of its own.\nFor example, the set $\\mathbb{Q}$ of rationals is not connected in $\\mathbb{R}$: the sets $(-\\infty, \\sqrt{2})$ and $(\\sqrt{2}, \\infty)$ split it.\n",
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