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  "body": "\n## Definition\n\nA [[topological space|0CQHFPGJ45]] $X$ is a *Fréchet–Urysohn space* when the following holds.\nFor every $A \\subseteq X$ and every point $x$ in the [[closure|0QEAJB91GZ]] $\\overline{A}$, there is a sequence $(x_n)$ in $A$ that [[converges|0W4CN9E2W5]] to $x$.\n\n## Remarks\n\nA Fréchet–Urysohn space is [[sequential|WMB2MAEDX4]]: if $A$ is sequentially closed and $x \\in \\overline{A}$, a sequence in $A$ converges to $x$, so $x \\in A$.\nEvery subspace of a Fréchet–Urysohn space is Fréchet–Urysohn.\nA sequential space is Fréchet–Urysohn if and only if each of its subspaces is sequential.\n\nExample: every [[first-countable space|0CP5NDD21C]] is Fréchet–Urysohn.\nFor $x \\in \\overline{A}$ and a decreasing neighbourhood base $U_1 \\supseteq U_2 \\supseteq \\cdots$ at $x$, pick $x_n \\in A \\cap U_n$; then $x_n \\to x$.\n\nNon-example: let $X$ be an uncountable set whose closed sets are $X$ and the countable subsets, and let $A \\subset X$ with $A$ and $X \\setminus A$ uncountable.\nThen $\\overline{A} = X$, but the only convergent sequences are eventually constant.\nSo a point $x \\notin A$ is in $\\overline{A}$ but is not the limit of a sequence in $A$.\n\nThe name Fréchet space is also used, but it has other meanings.\n",
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