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  "body": "\n## Definition\n\nLet $X$ be a [[topological space|0CQHFPGJ45]].\nThe space $X$ is a *Lindelöf space* if every [[open cover|049PXYHW1T]] $\\mathcal{U}$ of $X$ has a countable subcover.\nThat is, there is a countable family $\\mathcal{V} \\subseteq \\mathcal{U}$ with $\\bigcup \\mathcal{V} = X$.\n\n## Remarks\n\nExample: $\\R$ with the usual topology is Lindelöf.\nIt is [[second-countable|03PPNCM9GG]], with the countable base of open intervals with rational end points.\nFor an open cover $\\mathcal{U}$, pick for each basic interval contained in some member of $\\mathcal{U}$ one such member; these countably many members cover $\\R$.\n\nNon-example: an uncountable set $X$ with the discrete topology is not Lindelöf.\nThe cover $\\set{\\set{x} : x \\in X}$ is an open cover, and every subcover must contain all of its uncountably many members.\n\nThe [[Sorgenfrey line|Y4T3GVK8GV]] is Lindelöf, but its square, the Sorgenfrey plane, is not.\nSo a product of two Lindelöf spaces need not be Lindelöf.\n",
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