Compact Subspace of Linearly Ordered Space/Reverse Implication/Proof 1

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Theorem

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

Let $Y \subseteq X$ be a non-empty subset of $X$.


Let the following hold:

$(1): \quad$ For every non-empty $S \subseteq Y$, $S$ has a supremum and an infimum in $X$.
$(2): \quad$ For every non-empty $S \subseteq Y$: $\sup S, \inf S \in Y$.


Then $Y$ is a compact subspace of $\left({X, \tau}\right)$.


Proof

Let $\tau'$ be the $\tau$-relative subspace topology on $Y$.

Let $\preceq'$ be the restriction of $\preceq$ to $Y$.


Lemma

$\left({Y, \preceq', \tau'}\right)$ is a linearly ordered space.


Proof

By definition of a generalized ordered space:

$\tau'$ has a sub-basis consisting of upper and lower sets in $Y$.

To prove that $\left({Y, \preceq', \tau'}\right)$ is a linearly ordered space, we need to show that each element of this sub-basis is open in the $\preceq'$ order topology.

Let $U$ be a non-empty, $\tau'$-open upper set in $Y$.

If $U = \varnothing$ or $U = Y$, then $U$ is open in the $\preceq'$-order topology by the definition of topology.

Suppose then that $\varnothing \subsetneqq U \subsetneqq Y$.

Let $C = Y \setminus U$.

Then $C$ is non-empty and $\tau'$-closed.

$Y$ is $\tau'$-closed by Closed Set in Linearly Ordered Space.

Thus $C$ is also $\tau'$-closed.

$C$ has a supremum in $X$ by the premise.

Let $c = \sup_X C$

Since $C$ is $\tau'$-closed, Closed Set in Linearly Ordered Space shows that $c \in C$.

Since $c \in C = Y \setminus U$ and $U$ is an upper set in $Y$:

$c \prec u$ for all $u \in U$.

Furthermore, if $y \in Y$ and $c \prec y$, then by the definition of supremum, $y \notin C$, so $y \in U$.

Thus:

$U = {c^\succeq}_Y$

where ${c^\succeq}_Y$ denotes the upper closure of $c$ in $Y$.

So $U$ is open in the $\preceq'$-order topology for $Y$.

The same argument shows that $\tau'$-open lower sets in $Y$ are open in the $\preceq'$-order topology.

Thus $\tau'$ is the $\preceq'$-order topology.

$\Box$


The premises immediately show that $\left({Y, \preceq'}\right)$ is a complete lattice.

By Complete Linearly Ordered Space is Compact, $Y$ is a compact subspace of $X$.

$\blacksquare$