Vtyjkyuk

If $(X,d)$ is a metric space, then we know the topology generated by the set $$ B_d (x,delta) : text$x in X$ and $delta > 0$ $$ is called the metric topology.

Then how do you call the coarsest topology on $X times X$ such that the function $d:X times X to mathbb R$ is a continuous function?

Is there any relationship between those two topologies?

I don't think there is a specific name for the coarsest topology on $X times X$ for which $d : X times X to mathbbR$ is continuous. In general if $f$ is a function from a set $Y$ to a topological space $Z$, the coarsest topology on $Y$ for which $f$ is continuous is called the topology on $X$ generated by $f$. Therefore we may call it the topology on $X times X$ generated by $d$.

For the remainder let's fix the following notations:

• $mathcalO_d$ is the topology on $X times X$ generated by $d$;


• $mathcalO_mathrmmd$ is the metric topology on $X$;


• $mathcalO_mathrmpd sim mathcalO_mathrmmd otimes mathcalO_mathrmmd$ is the "metric product topology" on $X times X$.

$mathcalO_d$ is generated by the sets of the form $$d^-1 [ ( a , b ) ] = ( x , y ) in X times X : a < d (x,y) < b $$

Note that since $mathcalO_d$ is a topology on $X times X$ and $mathcalO_mathrmmd$ is a topology on $X$, the two topologies cannot be compared. We can, however, compare $mathcalO_d$ and $mathcalO_mathrmpd$. Of course, $mathcalO_d$ is coarser than $mathcalO_mathrmpd$, since $d$ is continuous with respect to $mathcalO_mathrmpd$.

These two topologies will differ greatly (except in trivial cases).

For instance, if $X$ has at least two points, then $mathcalO_d$ is not Hausdorff (it's not even T₀). This is because for all $x , y in X$ there is no open set containing exactly one of $(x,x)$ or $(y,y)$. On the other hand, $mathcalO_mathrmpd$ is always Hausdorff (even perfectly normal, since it is itself a metrizable topology).