The Sorgenfrey plane is the product topology on two copies of the Sorgenfrey line. It is separable because meets every nonempty basic rectangle . Its antidiagonal is an uncountable discrete subspace: meets it only at . 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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The Sorgenfrey plane is a topological space that is constructed from the real numbers, specifically using the Sorgenfrey line as its foundational element. The Sorgenfrey line is obtained by equipping the set of real numbers \(\mathbb{R}\) with a topology generated by half-open intervals of the form \([a, b)\), where \(a < b\). This creates a topology that is finer than the standard topology on \(\mathbb{R}\).