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  "body": "\n## Definition\n\nLet $(X, d)$ be a [[metric space|063W5JMY85]].\nThe metric topology of $(X, d)$ is the set $\\tau_d$ of all $U \\subseteq X$ such that for each $x \\in U$ there is a real $\\varepsilon \u003e 0$ with $B(x, \\varepsilon) \\subseteq U$.\nHere $B(x, \\varepsilon)$ is the [[open ball|0DC3MPR7C9]] with centre $x$ and radius $\\varepsilon$.\n\n## Remarks\n\nThe set $\\tau_d$ is a [[topology|0CNQXVBHCX]] on $X$: it contains $\\varnothing$ and $X$, and it is closed under [[unions|0HEF46RPEK]] and [[finite intersections|09NDE9V7X1]].\nEach open ball is in $\\tau_d$, by [[open balls are open|0B2305GNAZ]]. The open balls form a [[basis|06EX0FHBTF]] of $\\tau_d$.\nFor $\\mathbb{R}$ with the usual metric, $\\tau_d$ is the usual topology, generated by the open intervals.\nDifferent metrics can give the same topology. Such metrics are [[equivalent|0TWQ5PRKWS]].\n\n## Explanations\n\n### L1 human\n\nThe metric topology is how a metric space becomes a topological space.\nA set is open when each of its points has a small ball about it inside the set.\n",
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