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  "body": "\n## Statement\n\nLet $(X, d)$ be a [[metric space|063W5JMY85]], let $n \\ge 1$ be an integer, and let $U_1, \\ldots, U_n$ be [[open subsets|01ZQQDCERP]] of $X$.\nThen the [[intersection|0HBRW4E6G1]] $U_1 \\cap \\cdots \\cap U_n$ is open in $X$.\n\n## Remarks\n\nProof: let $x$ be in the intersection. For each $k$ there is $\\varepsilon_k \u003e 0$ with $B(x, \\varepsilon_k) \\subseteq U_k$. Then $\\varepsilon = \\min(\\varepsilon_1, \\ldots, \\varepsilon_n) \u003e 0$ and $B(x, \\varepsilon)$ lies in each $U_k$.\nThe statement is false for infinite families: in $\\mathbb{R}$, the intersection of the open intervals $(-1/n, 1/n)$ for $n \\ge 1$ is $\\{0\\}$, which is not open.\nIn a general [[topological space|0CNQXVBHCX]], this property is one of the axioms.\n\n## Explanations\n\n### L1 human\n\nEach open set gives its own small ball about the point, and the smallest of these finitely many balls fits in all the sets.\nWith infinitely many sets, the radii can shrink to zero, and then no ball is left.\n",
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