If $(X,d)$ is a metric space and $A$ and $B$ are subsets of $X$, then $d(A,B) = inf_{a\in A,b\in B} d(a,b)$. Now it is possible for $d(A,B)$ to be zero even if $A$ and $B$ are disjoint, or even if their closures are disjoint.
My question is, is the concept of two sets being zero distance apart a topological property? That is to say, if $d_1$ and $d_2$ induce the same topology on $X$, then is it necessarily true that $d_1(A,B)=0$ if and only if $d_2(A,B)$?
Let $X=\Bbb R^2$ be the Euclidean plane. Let $A$ be the $x$-axis, and $B$ the graph of $y=e^x$. Then in the usual distance, $d(A,B)=0$.
Define a homeomorphism $X\to X$ by $F(x,y)=(x,e^{-x}y)$. Then $F(A)=A$ and $F(B)$ is the line $y=1$, so $d(F(A),F(B))=1$.
So we can take $d_1$ to be the Euclidean metric, and $d_2(a,b) =d_1(F(a),F(b))$. These both give the standard topology, but $d_1(A,B)=0<d_2(A,B)$.