 Distance to a set

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 $d(x,F)\le d(x,z)+d(z,F)$; swapping $x,z$ proves $|d(x,F)-d(z,F)|\le d(x,z)$. 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.