 Closed diagonal of a metric space

For a metric space with the same metric on both factors, the diagonal is a closed subset of its product. At an off-diagonal point $(x,y)$ let $\delta =d(x,y)>0$. A maximum-metric ball of radius $\delta /3$ misses the diagonal, since a common coordinate $z$ would give $d(x,y)\le d(x,z)+d(z,y)<2\delta /3$. Equality of underlying sets without equality or compatibility of the metrics is a different assertion.