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  "body": "\n## Definition\n\nLet $X$ be a [[topological space|0CNQXVBHCX]] and let $S \\subseteq X$.\nThe boundary of $S$ is the [[set difference|0DT0TCV75N]]\n$$\\partial S = \\overline{S} \\setminus \\operatorname{int} S,$$\nwhere $\\overline{S}$ is the [[closure|0DRKT1QJR7]] and $\\operatorname{int} S$ is the [[interior|0C38S0KMJK]] of $S$.\n\n## Remarks\n\nA point $x$ is in $\\partial S$ if and only if each open set that contains $x$ meets both $S$ and $X \\setminus S$.\nThe boundary is closed, and $\\partial S = \\partial (X \\setminus S)$.\nIn $\\mathbb{R}$, the boundary of $(0, 1)$ is $\\{0, 1\\}$, and the boundary of $\\mathbb{Q}$ is $\\mathbb{R}$.\nAnother name for the boundary is the frontier.\n\n## Explanations\n\n### L1 human\n\nThe boundary is the edge of a set. It has the points where you can step into the set and also out of it with a very small step.\nFor a disc in the plane, the boundary is its circle.\n",
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