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  "body": "\n## Definition\n\nLet $X$ be a [[topological space|0CQHFPGJ45]].\nAn open cover $\\mathcal{V}$ of $X$ is a *refinement* of an open cover $\\mathcal{U}$ if every $V \\in \\mathcal{V}$ is contained in some $U \\in \\mathcal{U}$.\nThe space $X$ is *paracompact* if every [[open cover|049PXYHW1T]] of $X$ has an open refinement that is a [[locally finite family|DZ297MMGV8]].\n\n## Remarks\n\nExample: $\\R$ is paracompact.\nGiven an open cover $\\mathcal{U}$, each compact interval $[n, n + 1]$ is covered by finitely many members of $\\mathcal{U}$.\nIntersect these members with $(n - 1, n + 2)$; the resulting sets, over all $n \\in \\Z$, form a locally finite open refinement.\n\nNon-example: the [[long line|1DKCNS63DH]] is not paracompact, although it is Hausdorff and locally homeomorphic to $\\R$.\n\nEvery [[compact space|0TVJZYB6SH]] is paracompact, since a finite subcover is a locally finite refinement.\nSome authors include the Hausdorff condition in the definition of a paracompact space.\n",
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