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.

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Sources

• 1994: Martin J. Osborne and Ariel Rubinstein: A Course in Game Theory ... (previous) ... (next): $1.7$: Terminology and Notation