Sorgenfrey plane

Remarks

The basic open sets are the half-open rectangles $[a, b) \times [c, d)$. The anti-diagonal $D = \set{(x, -x) : x \in \R}$ is closed, since it is closed in the coarser Euclidean topology. It is discrete: the rectangle $[x, x + 1) \times [-x, -x + 1)$ meets $D$ only in $(x, -x)$. The plane is separable, because $\Q \times \Q$ is dense. By Jones' lemma, a separable normal space has no closed discrete subset $D$ with $2^{\abs{D}} > 2^{\aleph_0}$. Here $\abs{D} = 2^{\aleph_0}$, so the plane is not normal. Concretely, the points of $D$ with rational first coordinate and those with irrational first coordinate form two disjoint closed sets that no disjoint open sets separate. The plane also shows that a product of two Lindelöf spaces need not be Lindelöf.