open interval in $I_S$. $I_S$ has no isolated points because $0,1\notin S$. The double arrow space is obtained by splitting all points other than $0,1$. $I_\emptyset \cong I$, and $I_{{{\mathbb Q}}\cap (0,1)}$ is homeomorphic to the Cantor set.

For each $S \subseteq (0,1)$, $I_S$ is a compact separable LOTS with no isolated points. $I_S$ is second countable iff $S$ is countable. Every $I_\Lambda$ is a compact first countable LOTS.

$I_\Lambda$ will not be separable unless $\{x : \Lambda(x) > 1\}$ is countable. The study of compact separable LOTSes can be reduced to spaces of the form $I_S$. First note, by Lutzer and Bennett [@LB]:

\[lemma-hs\] If $X$ is a separable LOTS, then $X$ is hereditarily separable and hereditarily Lindelöf.

Also, it is easy to check:

\[lemma-rel\] If $X$ is a LOTS and $H$ is a compact subset of $X$, then the relative topology and the order topology agree on $H$.

Relating this to our $I_S$:

\[lemma-LOTS-standard\] Let $X$ be a compact separable LOTS. Then

-   If $X$ is perfect, then $X$ is homeomorphic to $I_S$ for some $S \subseteq (0,1)$.

-   If $X$ is not second countable, then $X$ has a closed subspace which is homeomorphic to $I_S$ for some 