Lower limit topology

In mathematics, the lower limit topology or right half-open interval topology is a topology defined on the set R {displaystyle mathbb {R} } of real numbers; it is different from the standard topology on R {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. In mathematics, the lower limit topology or right half-open interval topology is a topology defined on the set R {displaystyle mathbb {R} } of real numbers; it is different from the standard topology on R {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 R l {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 R l {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.