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Def/Total order

Definition of Totally ordered: Suppose that $X$ is a set, and $\leq$ is a partial order on $X$. Then, $\leq$ is a total order, or in other words, $(X, \leq)$ is a totally ordered set, if: $$\forall x,y \in X, x \leq y \mbox{ or } y \leq x.$$

Logical Connections

This definition logically relies on the following definitions and statements: Def/Partial order

The following statements and definitions logically rely on the material of this page: Def/Dedekind complete, Def/Dedekind cut, Def/Maximum, Def/Ordered ring, Def/Well ordered, State/N is totally ordered, and State/Squares are nonnegative in an ordered field

To visualize the logical connections between this definition and other items of mathematical knowledge, you can visit any of the following clusters, and click the "Visualize" tab: Clust/Ordered sets, Clust/Relations