Lower limit topology

Definition
Suppose $$X$$ is a linearly ordered set with the strict ordering denoted by $$<$$. The lower limit topology on $$X$$ is defined as the topology with the following basis: for $$a < b$$ in $$X$$, we have the basis element:

$$[a,b) = \{ x \mid a \le x < b \}$$

This topology is in general a finer topology than the order topology, though they coincide if every point has a predecessor.

The standard example of the lower limit topology is taking it on the real line, and the corresponding topological space is termed the Sorgenfrey line.