Closed Set in Linearly Ordered Space

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Theorem

Let $\left({X, \preceq, \tau}\right)$ be a linearly ordered space.

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


Then $C$ is closed in $X$ iff for all non-empty subsets $S$ of $C$:

If $s \in X$ is a supremum or infimum of $S$ in $X$, then $s \in C$.


Proof

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 $\left({{a}\,.\,.\,{q}}\right)$ 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 $S \subseteq C$, ${\uparrow}a \cap S = \varnothing$, 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.

$\Box$


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 $\left({a \,.\,.\, b}\right)$ 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.

$\blacksquare$


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