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  "body": "\n## Summary\n\nThe topology of uniform convergence is the topology on a set of functions into a metric space in which convergence means [[uniform convergence|08VZ57PT26]].\nIt comes from the uniform metric, which measures the largest distance between the values of two functions.\nIt is finer than the [[topology of pointwise convergence|FVGB8ZP67B]].\nOn continuous maps, uniform limits stay continuous, which is why this topology is so useful.\n\n## Explanations\n\n### L1 human\n\nTwo functions are close in this topology when their graphs stay close everywhere at once.\nA band of fixed width around the graph of $f$ must contain the whole graph of $g$.\nFor example, $f_n(x) = x / n$ on $[0, 1]$ converges uniformly to $0$, since $\\abs{f_n(x)} \\le 1/n$ for all $x$.\nBut $f_n(x) = x^n$ on $[0, 1)$ does not, because near $x = 1$ the values stay close to $1$.\n",
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