= Distance to a set
{title2=$d(x,F)=\inf_{y\in F}d(x,y)$}
= Distances to a set
{synonym}
For a nonempty <subset> $F$ of a <metric space>, the <distance to a set> is $d(x,F)=\inf_{y\in F}d(x,y)$. The <triangle inequality> gives $d(x,F)\leq d(x,z)+d(z,F)$; swapping $x,z$ proves $|d(x,F)-d(z,F)|\leq d(x,z)$. Thus it is one-<Lipschitz>. Its zero set is exactly the <closure> of $F$: every <neighbourhood> of $x$ meets $F$ precisely when this infimum vanishes. For a <closed set>, zero distance therefore implies membership. When the empty set is allowed, use the convention $d(x,\varnothing)=+\infty$; the finite-valued Lipschitz assertion applies to nonempty sets.
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