Maximum product metric ($D((x,x^{\prime }),(y,y^{\prime }))=\max \{d(x,y),d^{\prime }(x^{\prime },y^{\prime })\}$)

The displayed metric on a product of two metric spaces satisfies its triangle inequality because $\max \{a+c,b+d\}\le \max \{a,b\}+\max \{c,d\}$ for nonnegative coordinates. Its open balls are products of equal-radius balls, so it induces the product topology.

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.