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Let $\Omega$ be a totally-ordered set.

We need to introduce some symbolism and terminology, which is all very natural and intuitive. If $A$ and $B$ are subsets of $\Omega$ we write $A<B$ whenever $a<b$ for all $a\in A$ and $b\in B$. By an interval of $\Omega$, we mean any subset $I$ such that there exist sets $A$ and $B$ with $A<I<B$ and $\Omega=A\cup I\cup B$. In the special case where $A=\emptyset$, the interval $I$ is called an initial segment of $\Omega$.

It is desirable to write intervals of $\Omega$ as we might write intervals of $\mathbb{R}$. Thus, we say that an interval $I$ of $\Omega$ is of real form if it can be written in one of the following ways for some $a,b\in\Omega$: $$(-\infty,b):=\{x\in\Omega:x<b\}$$ $$(-\infty,b]:=\{x\in\Omega:x\leq b\}$$ $$(a,b):=\{x\in\Omega:a<x<b\}$$ and so on, also for $[a,b)$, $(a,b]$, $[a,b]$, $(a,\infty)$, and $[a,\infty)$, defined in the obvious ways.

Unfortunately, intervals of $\Omega$ need not always have real form. Consider for instance the case $\Omega=[0,1)\cup(1,2]$ endowed with the usual ordering. Then $[0,1)$ is an interval, but $1\notin\Omega$ and so it can't be written in real form. On the other hand, due to the fact that $\Omega\subseteq[0,2]$ in this case, we can still write the interval $[0,1)$ very nicely.

Fact. For any totally-ordered set $\Omega$, there exists a superset $\Gamma\supseteq\Omega$ which is endowed with a "compatible" total order (in the sense that $x<y$ in $\Omega$ implies $x<y$ in $\Gamma$), and such that every interval of $\Gamma$ has real form.

This is not hard to show. If $b\in\Omega$ we define $$I_b:=(-\infty,b].$$ Let $\Gamma$ denote the set of all initial segments of $\Omega$. Then $\subseteq$ is a total order on $\Gamma$ compatible with the order on $\Omega$, and the map $b\mapsto I_b$ is an injection from $\Omega$ into $\Gamma$ which means we can identify $\Omega$ with a subset of $\Gamma$. It is straightforward to see that every interval of $\Gamma$ is of real form.

Question 1. Has this been discussed in the literature? If so, does anyone have a reference?

Question 2. The term "real form" sounds unsatisfying, but I can't think of anything better. Is there already a term for this in the literature? Or just a better suggestion from you guys?

Thanks!

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  • $\begingroup$ A general term for interval is order convex which can also be used with partial orders. $\endgroup$ – William Elliot Feb 21 at 23:54
  • $\begingroup$ As a matter of fact, a set in a connected linear order is order convex iff it is an interval. So the real forms are the connected orders. $\endgroup$ – William Elliot Feb 22 at 10:18
  • $\begingroup$ @WilliamElliot connected in the order topology? $\endgroup$ – Ben W Feb 22 at 14:03
  • $\begingroup$ Yes............ $\endgroup$ – William Elliot Feb 22 at 22:06

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