Closed Set in Linearly Ordered Space

Theorem
Let $\struct {X, \preceq, \tau}$ be a linearly ordered space.

Let $C$ be a subset of $X$.

Then $C$ is closed in $X$ for all non-empty subsets $S$ of $C$:
 * If $s \in X$ is a supremum or infimum of $S$ in $X$, then $s \in C$.

Necessary Condition
Suppose that $C$ be closed.

Let $S$ be a non-empty subset of $C$.

Let $b \in X \setminus C$.

We will show that $b$ is not a supremum of $S$.

If $b$ is not an upper bound of $S$, then by definition $b$ cannot be a supremum of $S$.

Suppose, then, that $b$ is an upper bound of $S$.

Since $C$ is closed and $b \notin C$, there must be an open interval or open ray $U$ containing $b$ that is disjoint from $Y$.

Since $b$ is an upper bound for $S$ and $S$ is not empty, $U$ cannot be a downward-pointing ray.

Thus $U$ is either an open interval $\openint a q$ or an upward-pointing open ray ${\uparrow}a$.

Then $a \prec b$.

Since $b \in U$, and $b$ is an upper bound of $S$, no element strictly succeeding all elements of $U$ can be in $S$.

By the above and the fact that $S \subseteq C$, ${\uparrow}a \cap S = \O$, so $a$ is an upper bound of $S$.

Since $a \prec b$, $b$ is not a supremum of $S$.

A similar argument shows that $b$ is not an infimum.

Sufficient Condition
Suppose that no nonempty subset of $C$ has a supremum or infimum relative to $X$ that lies in $X \setminus C$.

Let $p \in X \setminus C$.

Case 1: $p$ is an upper bound of $C$
Since $C$ is a non-empty subset of $C$, it does not have a supremum in $X \setminus C$.

Thus $p$ is not a supremum of $C$.

Therefore, $C$ has an upper bound $a \in X$ such that $a \prec p$.

Thus ${\uparrow_X}a$ contains $p$ and is disjoint from $C$, so $p$ is not an adherent point of $C$.

Case 2: $p$ is a lower bound of $C$
The same approach used for case 1 proves $p$ is not an adherent point of $C$.

Case 3: $p$ is neither an upper nor a lower bound of $C$
In this case, $C \cap {\downarrow_X} p$ and $C \cap {\uparrow_X} p$ are nonempty, and their union is $C$.

Thus $p$ is not a supremum of $C \cap {\downarrow_X} p$ and is not an infimum of $C \cap {\uparrow_X}p$.

Thus there are $a, b \in X$ such that $a \prec p \prec b$, $a$ is an upper bound of $C \cap {\downarrow_X}p$, and $b$ is a lower bound of $C \cap {\downarrow_X} p$.

Then $\openint a b$ contains $p$ and is disjoint from $C$, so $p$ is not an adherent point of $C$.

Since $C$ contains all of its adherent points, it is closed.

Note
Non-empty subsets of $C$ may not have suprema or infima in $X$, but if they have them, they are elements of $C$.

Also see

 * Compact Subspace of Linearly Ordered Space