uncountable $S \subseteq (0,1)$.

-   $X$ is homeomorphic to a subset of $I_S$ for some $S \subseteq (0,1)$.

For (1): Let $E \subseteq X$ be countable and dense in $X$ and contain the first and last elements of $X$. Let $B$ be the set of all $b \in E$ such that for some $a \in E$: $a < b$ and $(a,b) = \emptyset$. Let $D = E \backslash B$. Since $X$ has no isolated points, $D$ is also dense in $X$ and contains the first and last elements of $X$, and is also densely ordered. Let $f$ be an order isomorphism from $D$ onto ${{\mathbb Q}}\cap [0,1]$. Then $f$ extends in a natural way to a continuous $F: X {\twoheadrightarrow}[0,1]$, and $1 \le |F{^{-1}}\{r\}| \le 2$ for each $r \in [0,1]$. Let $S = \{r : |F{^{-1}}\{r\}| = 2\}$.

For (2): Since $X$ is hereditarily Lindelöf, the Cantor-Bendixson sequence of $X$ has countable length and removes countably many points. Thus, $X$ is not scattered, and, letting $H$ be the perfect kernel of $X$, $X \backslash H$ is countable. Then $H$ is separable and not second countable, so $H \cong I_S$ for some uncountable $S$.

For (3): Apply (1) to the 