Sorgenfrey plane

The Sorgenfrey plane $\mathbb{R}_l \times \mathbb{R}_l$ is a useful example in topology. It is formed by taking the Cartesian product of the lower-limit topology on $\mathbb{R}$ with itself.

Let $-\Delta = \{x \times (-x) \,|\, x \in \mathbb{R}_l\}$ be the anti-diagonal. It is a closed subspace of the Sorgenfrey plane, but it inherits the discrete topology as a subspace: consider the basis element given by $[x,x+1) \times [-x,-x+1)$.

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Countability axioms

The Sorgenfrey plane is first-countable, is separable (has a countably dense subset, namely $\mathbb{Q}^2$), but not Lindelof, and consequently not second-countable (does not have a countable basis).

Separation axioms

The Sorgenfrey plane is regular, but not normal. It is regular because it is the Cartesian product of regular spaces. It is not normal; we can see this because any subset $A$ of $-\Delta$ is a closed subspace of $\mathbb{R}_l^2$, and it can be shown that there do not exist disjoint open sets about $A$ and $-\Delta \setminus A$ in $\mathbb{R}_l^2$.

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