Lower limit topology

In mathematics, the lower limit topology or right half-open interval topology is a topology defined on ${\displaystyle \mathbb {R} }$, the set of real numbers; it is different from the standard topology on ${\displaystyle \mathbb {R} }$ (generated by the open intervals) and has a number of interesting properties. It is the topology generated by the basis of all half-open intervals [a,b), where a and b are real numbers.

The resulting topological space is called the Sorgenfrey line after Robert Sorgenfrey or the arrow and is sometimes written ${\displaystyle \mathbb {R} _{l}}$. Like the Cantor set and the long line, the Sorgenfrey line often serves as a useful counterexample to many otherwise plausible-sounding conjectures in general topology. The product of ${\displaystyle \mathbb {R} _{l}}$ with itself is also a useful counterexample, known as the Sorgenfrey plane.

In complete analogy, one can also define the upper limit topology, or left half-open interval topology.