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  "body": "\n## Definition\n\nLet $X$ be a set and let $\\tau_1$ and $\\tau_2$ be [[topologies|0CNQXVBHCX]] on $X$.\nIf $\\tau_1 \\subseteq \\tau_2$, then $\\tau_1$ is coarser (or weaker) than $\\tau_2$, and $\\tau_2$ is finer (or stronger) than $\\tau_1$.\nIf also $\\tau_1 \\neq \\tau_2$, then $\\tau_1$ is strictly coarser and $\\tau_2$ is strictly finer.\n\n## Remarks\n\nOn each set $X$, the [[indiscrete topology|0XJ02JBD6S]] is the coarsest topology and the [[discrete topology|0S2TEHX4E1]] is the finest topology.\nWe have $\\tau_1 \\subseteq \\tau_2$ if and only if the identity map $(X, \\tau_2) \\to (X, \\tau_1)$ is [[continuous|0SY9ECQ19Q]].\nTwo topologies need not be comparable: on $\\{a, b\\}$, the topologies $\\{\\varnothing, \\{a\\}, X\\}$ and $\\{\\varnothing, \\{b\\}, X\\}$ are not comparable.\n\n## Explanations\n\n### L1 human\n\nA finer topology has more open sets, so it can separate points in more ways.\nA coarser topology has fewer open sets, so more sequences converge and more maps into it are continuous.\n",
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