= Sorgenfrey plane
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The Sorgenfrey plane is the <product topology> on two copies of the <Sorgenfrey line>. It is separable because $\mathbb Q^2$ meets every nonempty basic rectangle $[a,b)\times[c,d)$. Its antidiagonal $\{(x,-x):x\in\mathbb R\}$ is an uncountable discrete subspace: $[x,x+\epsilon)\times[-x,-x+\epsilon)$ meets it only at $(x,-x)$. In a discrete space every dense subset is the whole space, so this subspace is not separable. Thus separability need not pass to subspaces of nonmetrizable spaces.
// Target: analysis.bigb
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