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  "body": "\n## Definition\n\nLet $(X, d)$ be a [[metric space|063W5JMY85]] and let $S \\subseteq X$.\nThe set $S$ is totally bounded when for each real $\\varepsilon \u003e 0$ there are finitely many sets $A_1, \\ldots, A_n \\subseteq X$ with $S \\subseteq A_1 \\cup \\cdots \\cup A_n$ and $\\operatorname{diam}(A_k) \u003c \\varepsilon$ for each $k$.\nThe space $X$ is totally bounded when $X$ itself is a totally bounded set.\n\n## Remarks\n\nHere $\\operatorname{diam}$ is the [[diameter|0692CGQF00]].\nEach totally bounded set is [[bounded|0P953MG05E]]. The converse is false: an infinite set with the [[discrete metric|0NA4FK7B8S]] is bounded but not totally bounded.\nIn $\\mathbb{R}^n$ with the Euclidean metric, a set is totally bounded if and only if it is bounded.\nThe [[epsilon-net characterisation|0NMGJJ19HD]] gives an equivalent form with balls.\n\n## Explanations\n\n### L1 human\n\nA totally bounded set can be cut into finitely many small pieces, for each size of piece that you choose.\nA bounded set only fits in one big ball, and this is not enough in infinite dimensions.\n",
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