Definition:Continuous Total Preordering

Definition
Let $S$ be a set.

Let $\precsim$ be a total preordering on $S$.

Let $\precsim$ be such that:
 * $a \precsim b$ whenever there exist sequences $\left\langle{a^k}\right\rangle_k$ and $\left\langle{b^k}\right\rangle_k$ that converge to $a$ and $b$ respectively for which $a^k \precsim b^k$ for all $k$.

Then $\precsim$ is continuous.