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  "body": "\n## Definition\n\nLet $X$ be a set and let $\\mathcal{S}$ be a set of subsets of $X$.\nThe topology generated by $\\mathcal{S}$ is the smallest [[topology|0CNQXVBHCX]] on $X$ that contains $\\mathcal{S}$. It is the [[intersection|0839XNQXDK]] of all topologies on $X$ that contain $\\mathcal{S}$.\nIf $(X, \\tau)$ is a topological space and $\\tau$ is the topology generated by $\\mathcal{S}$, then $\\mathcal{S}$ is a subbasis of $\\tau$.\n\n## Remarks\n\nThe open sets of the generated topology are the unions of finite intersections of members of $\\mathcal{S}$. Here the empty intersection is $X$.\nThus the finite intersections of members of $\\mathcal{S}$ form a [[basis|06EX0FHBTF]] of the generated topology.\nThe open rays $(-\\infty, a)$ and $(b, \\infty)$, with $a, b \\in \\mathbb{R}$, form a subbasis of the usual topology of $\\mathbb{R}$.\nThe sets $\\pi_i^{-1}(U)$, with $U$ open, form a subbasis of the [[product topology|04S9HX3GAP]].\n\n## Explanations\n\n### L1 human\n\nA subbasis is a set of open sets that you want to have. The topology is the smallest one that makes them all open.\nYou get all open sets by taking finite intersections of them and then arbitrary unions.\n",
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