Sorgenfrey

The Sorgenfrey line is the real line equipped with the lower limit topology, obtained by taking as a basis all half-open intervals [a, b) with a < b. This topology is strictly finer than the standard Euclidean topology on the real numbers, so more sets are open. The space is T1, regular, and first-countable, and it is separable since the rational numbers are dense in it. However, it is not metrizable and not second countable, illustrating that separability and first-countability do not imply metrizability or second countability.

The topology is named after Lilian M. Sorgenfrey, who introduced it in the 1950s as a tool in general topology. It is a standard example used to investigate how properties behave under change of topology and, in particular, how product spaces can fail to inherit certain niceties from their factors.

A closely related construction is the Sorgenfrey plane, the product of two Sorgenfrey lines. The Sorgenfrey plane is not normal, providing a classical counterexample in topology to show that the product of normal spaces need not be normal. The Sorgenfrey plane exhibits several pathological features and is widely discussed in the study of product spaces, topology bases, and separation axioms.

Together, the Sorgenfrey line and plane serve as fundamental examples in point-set topology, illustrating the distinct behavior of topological properties under refinement of a topology and under taking products.