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  "body": "\n## Proof\n\n1. Let $(X, \\tau)$ be a [[topological space|0CNQXVBHCX]] and let $S \\subseteq X$.\n2. Let $\\mathcal{U}$ be the family of all [[open sets|01ZQQDCERP]] $U$ of $X$ with $U \\subseteq S$. Each member of $\\mathcal{U}$ is in $\\tau$.\n3. By the definition of the [[interior|0C38S0KMJK]], $\\operatorname{int} S$ is the [[union|05GK32H72H]] of the members of $\\mathcal{U}$.\n4. By the second axiom of a topology, the union of each family of members of $\\tau$ is in $\\tau$.\n5. So $\\operatorname{int} S \\in \\tau$, that is, $\\operatorname{int} S$ is open in $X$.\n\n## Idea\n\nThe interior is built as a union of open sets. The axioms of a topology say that every such union is open. So no work is needed beyond the definition.\n",
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