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  "body": "\n## Definition\n\nLet $X$ be a [[topological space|0CQHFPGJ45]].\nFor a family $\\mathcal{G}$ of subsets of $X$ and a point $x \\in X$, the *star* of $x$ in $\\mathcal{G}$ is $\\operatorname{st}(x, \\mathcal{G}) = \\bigcup \\set{G \\in \\mathcal{G} : x \\in G}$.\nA *development* of $X$ is a sequence $\\mathcal{G}_1, \\mathcal{G}_2, \\dots$ of [[open covers|049PXYHW1T]] of $X$ with the following property.\nFor every $x \\in X$ and every open set $U$ with $x \\in U$, there is $n$ with $\\operatorname{st}(x, \\mathcal{G}_n) \\subseteq U$.\nThe space $X$ is *developable* if it has a development.\nA *Moore space* is a developable space that is [[regular|03HMJASQBQ]] and [[Hausdorff|0S1FJ8VEX9]].\n\n## Remarks\n\nEvery metric space $(X, d)$ is a Moore space.\nLet $\\mathcal{G}_n$ be the cover by all open balls of radius $1/n$.\nEvery ball of $\\mathcal{G}_n$ that contains $x$ lies in the ball $B(x, 2/n)$, so $\\operatorname{st}(x, \\mathcal{G}_n) \\subseteq B(x, 2/n)$, and this lies in $U$ for $n$ large.\nA set $X$ with two points and the indiscrete topology is developable, with $\\mathcal{G}_n = \\set{X}$ for all $n$, because $X$ is the only open set containing a point.\nIt is not a Moore space, because it is not Hausdorff.\nSome authors do not include regularity or the Hausdorff condition in the definition, so the word should be checked in each text.\n\n## Explanations\n\n### L1 human\n\nA development is a sequence of open covers that get finer and finer around every point, like the covers by balls of radius $1/n$ in a metric space.\nA Moore space is a well separated space with such a sequence.\n",
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  "name": "Moore space",
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