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  "body": "\n## Definition\n\nLet $X$ be a [[topological space|0CQHFPGJ45]].\nA family $\\mathcal{V}$ of subsets of $X$ is *point-finite* if each $x \\in X$ belongs to only finitely many members of $\\mathcal{V}$.\nThe space $X$ is *metacompact* if every [[open cover|049PXYHW1T]] $\\mathcal{U}$ of $X$ has a point-finite open refinement.\nThat is, there is a point-finite open cover $\\mathcal{V}$ such that each $V \\in \\mathcal{V}$ lies in some $U \\in \\mathcal{U}$.\n\n## Remarks\n\nExample: every [[paracompact space|8R29G6NG9Q]] is metacompact.\nA [[locally finite|DZ297MMGV8]] family is point-finite, since a neighbourhood of $x$ meeting finitely many members also shows that $x$ lies in finitely many.\n\nNon-example: the space $[0, \\omega_1)$ of countable ordinals with the order topology is not metacompact.\nIt is [[countably compact|S3N3CHZV89]] but not compact.\nA countably compact metacompact space is always compact.\n",
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