The Sorgenfrey line (after Robert Sorgenfrey?, a mathematical descendant of R.L. Moore?) is the real line$\mathbb{R}$, but topologized by taking as topological basis the collection of half-open intervals$[a, b)$. This is sometimes called the lower limit topology?, and the Sorgenfrey line is often denoted by $\mathbb{R}_l$.

The Sorgenfrey line and spaces derived from it, notably the Sorgenfrey plane$\mathbb{R}_l \times \mathbb{R}_l$ (with the product topology), is a rich source for counterexamples in general topology. For example, the Sorgenfrey line is a ($T_1$) normal space and even a paracompact space, but the Sorgenfrey plane fails to be normal (and thus is not paracompact either). The Sorgenfrey plane is also an example of a separable space, but admitting an uncountablediscretesubspace (which is thus not separable), showing that separability is not a hereditary property.

Revised on May 22, 2017 15:20:01
by Urs Schreiber
(92.218.150.85)