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  "body": "\n## Definition\n\nLet $X$ be a [[topological space|0CNQXVBHCX]].\nThe space $X$ is first-countable when each point $x \\in X$ has a sequence $(N_n)_{n \\ge 1}$ of [[neighbourhoods|09CJ00H3NW]] with this property: each neighbourhood of $x$ contains some $N_n$.\n\n## Remarks\n\nSuch a sequence is a [[countable|0ZG61TFBES]] neighbourhood basis at $x$. One can choose the sets $N_n$ open, and also decreasing.\nEach metric space is first-countable: take $N_n = B(x, 1/n)$, as in [[open balls form a basis|02EEX96DFS]].\nEach [[second-countable space|00RE1R4W7N]] is first-countable.\nIn a first-countable space, a point is in the closure of a set $A$ exactly when it is a limit of a sequence in $A$.\n\n## Explanations\n\n### L1 human\n\nAt each point, you can shrink to the point along a countable list of neighbourhoods.\nIn a metric space, the balls of radius $1/n$ do this.\n",
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