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  "body": "\n## Definition\n\nLet $X$ be a set and let $\\mathcal{M}$ be a family of subsets of $X$, that is, a subset of the [[power set|0BC7KX51Z8]] of $X$.\nThe family $\\mathcal{M}$ is a *Moore family* on $X$ when both conditions hold.\n\n1. $X \\in \\mathcal{M}$.\n2. For each nonempty subfamily $\\mathcal{N} \\subseteq \\mathcal{M}$, the [[intersection|0839XNQXDK]] $\\bigcap \\mathcal{N}$ lies in $\\mathcal{M}$.\n\n## Remarks\n\nThe closed sets of a topological space $X$ form a Moore family: $X$ is closed, and any intersection of closed sets is closed.\nThe open sets of $\\R$ do not: the open intervals $(-1/n, 1/n)$ for $n \\ge 1$ have intersection $\\set{0}$, which is not open.\nIf $\\mathcal{M}$ is a Moore family, then $c(A) = \\bigcap \\set{M \\in \\mathcal{M} : A \\subseteq M}$ is a [[closure operator|17FVAN6HXX]] on the subsets of $X$.\nIts closed sets are exactly the members of $\\mathcal{M}$.\nConversely, the closed sets of any closure operator on the subsets of $X$ form a Moore family.\nWith the convention that the empty intersection is $X$, the two conditions merge into one: $\\mathcal{M}$ is closed under all intersections.\nOther names are closure system and intersection-closed family.\n\n## Explanations\n\n### L1 human\n\nA Moore family is a collection of subsets that contains the whole set and survives taking intersections.\nIts members are the \"closed\" sets of a closure operator.\n",
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