Sparse linear orders are the contiguous subsets of the integers

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Order notions. For a (pre)ordered set $Q$: $Q$ is sparse if every closed interval $[a,b]=\{q: a\le q\le b\}$ is finite; $Q$ is dense (book sense) if $[a,b]$ is infinite whenever $a<b$; $A\subseteq Q$ is a dense subset if for all $q_1<q_2$ there is $q\in A$ with $q_1\le q\le q_2$; $Q$ is complete if every nonempty subset that is bounded above has a least upper bound in $Q$. A linear order $\preceq$ on $X$ is a natural order if the order topology of $\preceq$ (generated by all rays $\{x: a\prec x\}$ and $\{x: x\prec b\}$) equals the natural topology $\mathcal T$. $D$ is fully characterized by a topological space $Q$ if there is a homeomorphism $(X,\mathcal T)\to Q$. "Decidable" means $\Omega\setminus s\in\mathcal D$ for all $s\in\mathcal D$.

Statement. For every linearly ordered type $Q$: $Q$ is sparse (every interval $[a,b]$ finite) if and only if there is a set $S\subseteq\mathbb Z$ that is order-connected (whenever $a,b\in S$ and $a\le c\le b$ then $c\in S$) together with an order isomorphism $Q\cong S$ (with the order inherited from $\mathbb Z$). $S=\emptyset$ is allowed.