= Star metric centered at a point
Given a <metric space> $(X,d)$ and $a\in X$, define a new <metric> by $d_a(x,x)=0$ and $d_a(x,y)=d(x,a)+d(a,y)$ for distinct points. It is the distance obtained by joining every point to a central vertex with its original distance to $a$ as edge length. Symmetry and positive definiteness are immediate. With three distinct points the <triangle inequality> follows from the extra nonnegative term $2d(z,a)$ in the route through an intermediate point; repeated-point cases are immediate.
Every $b\ne a$ is isolated, since $B_{d_a}(b,d(b,a))=\{b\}$. At the centre, however, $B_{d_a}(a,r)=B_d(a,r)$. Thus a map out of this space is <continuous> exactly when it is continuous at $a$ in the original metric. The <closure> of $B$ is $B$ together with $a$ if $a\notin B$ and $d(a,B)=0$; otherwise it is simply $B$. This construction isolates every point except the centre while preserving exactly its original neighbourhood filter.
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