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:

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

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.