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  "body": "\n## Definition\n\nLet $X$ be a set.\nA *proximity* on $X$ is a relation $\\delta$ between subsets of $X$, written $A \\mathrel{\\delta} B$ and read \"$A$ is near $B$\", such that for all $A, B, C \\subseteq X$:\n\n1. If $A \\mathrel{\\delta} B$, then $B \\mathrel{\\delta} A$.\n2. If $A \\mathrel{\\delta} B$, then $A \\neq \\emptyset$.\n3. If $A \\cap B \\neq \\emptyset$, then $A \\mathrel{\\delta} B$.\n4. $A \\mathrel{\\delta} (B \\cup C)$ if and only if $A \\mathrel{\\delta} B$ or $A \\mathrel{\\delta} C$.\n5. If $A \\mathrel{\\delta} E$ or $B \\mathrel{\\delta} (X \\setminus E)$ holds for every $E \\subseteq X$, then $A \\mathrel{\\delta} B$.\n\nA *proximity space* is a pair $(X, \\delta)$ of a set $X$ and a proximity $\\delta$ on $X$.\n\n## Remarks\n\nLet $(X, d)$ be a [[metric space|0VKD1TP7F3]] and let $A \\mathrel{\\delta} B$ mean $\\inf \\set{d(a, b) : a \\in A, b \\in B} = 0$, with $\\inf \\emptyset = \\infty$.\nClauses 1 to 4 follow directly.\nFor clause 5, suppose that $A$ is not near $B$, so the distance $r$ between them is positive.\nLet $E = \\set{x : d(x, A) \\ge r/2}$.\nThen $A$ is not near $E$, and each point of $X \\setminus E$ is at distance more than $r/2$ from $B$, so $B$ is not near $X \\setminus E$.\nThe relation that makes all pairs of subsets near fails clause 2, since it makes $\\emptyset$ near $\\emptyset$.\nA proximity defines a topology: the closure of $A$ is the set of points $x$ with $\\set{x} \\mathrel{\\delta} A$.\nMany authors also ask that $\\set{x} \\mathrel{\\delta} \\set{y}$ imply $x = y$; such a proximity is *separated*.\n\n## Explanations\n\n### L1 human\n\nA proximity is a rule that says which pairs of sets touch or almost touch.\nThe clauses say that nearness is symmetric, that the empty set is near nothing, and that sets that meet are near.\nA set is near a union exactly when it is near one of the parts.\nThe last clause says that two sets that are not near can be kept apart by a set in between.\n",
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