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  "body": "\n## Proof\n\nWe unfold the definitions.\n\n1. Let $(X, d)$ be a [[metric space|063W5JMY85]] and let $U \\subseteq X$.\n2. By the definition of an [[open set|01ZQQDCERP]] of a metric space, $U$ is open in $X$ if and only if $U \\in \\tau_d$. Here $\\tau_d$ is the [[metric topology|0A6YRNZB2R]].\n3. By the definition of $\\tau_d$, we have $U \\in \\tau_d$ if and only if each $x \\in U$ has a real $\\varepsilon \u003e 0$ with $B(x, \\varepsilon) \\subseteq U$.\n4. Here $B(x, \\varepsilon)$ is the [[open ball|0DC3MPR7C9]] with centre $x$ and radius $\\varepsilon$.\n5. Steps 2 and 3 together give: $U$ is open if and only if each $x \\in U$ has an $\\varepsilon \u003e 0$ with $B(x, \\varepsilon) \\subseteq U$.\n\n## Idea\n\nThe open sets of a metric space are by definition the members of the metric topology. That topology is defined by the ball condition, so the statement only puts the two definitions side by side.\n",
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