= Closed diagonal of a metric space
{title2=$\Delta=\{(x,x):x\in X\}$}
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.
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