\begin{definition}[Definition:Left Order Topology]
Let $\struct {S, \preccurlyeq}$ be a totally ordered set.
Let $\tau$ be the topology on $S$ generated by the basis sets of the form:
:$S_a = \set {x: x \prec a}$
for $a \in S$.
Then the topological space $\struct {S, \preccurlyeq, \tau}$ is known as the '''left order topology on $S$'''.
\end{definition}