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  "body": "\n## Definition\n\nLet $(X, \\tau)$ be a [[topological space|0CNQXVBHCX]].\nA basis of $\\tau$ is a set $\\mathcal{B} \\subseteq \\tau$ such that each $U \\in \\tau$ is a [[union|05GK32H72H]] of members of $\\mathcal{B}$.\nEquivalently, $\\mathcal{B} \\subseteq \\tau$ and for each $U \\in \\tau$ and each $x \\in U$ there is $B \\in \\mathcal{B}$ with $x \\in B \\subseteq U$.\n\n## Remarks\n\nA family $\\mathcal{B}$ of subsets of a set $X$ is a basis of some topology on $X$ if and only if two conditions hold.\nFirst, the members of $\\mathcal{B}$ cover $X$. Second, for all $B_1, B_2 \\in \\mathcal{B}$ and $x \\in B_1 \\cap B_2$ there is $B_3 \\in \\mathcal{B}$ with $x \\in B_3 \\subseteq B_1 \\cap B_2$. This topology is then unique.\nThe open intervals form a basis of the usual topology of $\\mathbb{R}$. The intervals with rational endpoints form a countable basis.\nIn a metric space, the [[open balls form a basis|02EEX96DFS]] of the metric topology.\n\n## Explanations\n\n### L1 human\n\nA basis is a set of building blocks: each open set is a union of blocks.\nIn the plane, you can build each open set from open discs, or from open squares.\n",
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