Franklin's Notes

Interval of an ordering

In order theory, an interval of an ordered set $A$ is a subset $I\subset A$ such that for any two elements $x,y\in I$ with $z\in A$ such that $x<z<y$, it follows that $z\in I$ as well. A proper interval is a nonempty interval that does not consist of the entire ordering. Intervals are also sometimes called convex suborders. If $x,y$ are any two elements of $A$, then the following are all intervals of $A$:

However, it is not necessarily true that all intervals of $A$ take this form. This is true for some well-known orderings though, like the order types $\omega$ of $\mathbb N$, $\zeta$ of $\mathbb Z$, and $\lambda$ of $\mathbb R$. For this reason, there is a lot of potential for the term "interval" to be misinterpreted in order theory as referring to a subordering of one of the following $8$ forms, rather than the more general definition as a subset which is closed under taking intermediate values. As an example of an order type in which not every interval takes one of these forms, consider $\omega+\omega^\ast$, or $\eta$, the order type of $\mathbb Q$.



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