(a) If $f:[a,c)\to[0,1)$ is a bijection preserving the order, then $f:[a,b)\to[0,f(b))$ is a bijection preserving the order, similar for $[b,c)$ . Vice versa, if there are two order preserving bijections from $[a,b)$ and $[b,c)$ to $[0,1/2)$ and $[1/2,1)$ then the joint function is the order preserving bijection needed.(b) $x_0<x_i<x_{i+1}<b$ and $[x_0,b)$ has the order type of $[0,1)$ implies, by applying (a) two times, that $[x_i,x_{i+1})$ has the order type of $[0,1)$ . Vice versa, we construct bijective order preserving functions $[x_i,x_{i+1})\to[1-1/2^i,1-1/2^{i+1})$ and the resulting function is an order preserving surjective function onto $[0,1)$ .(c) Transfinite induction: we show that if it holds for any $x\in S_a-\{a_0\}$ then it holds for $a$ . If $a$ has an immediate predecessor $b$ then $[a_0\times 0,b\times 0)$ is either empty or has the order type of $[0,1)$ , and $[b\times 0,a\times 0)$ has it as well. By (a), the property holds for $a$ . If $a$ has no immediate predecessor, then the set of all elements lower than $a$ has it as the supremum (for any $b<a$ there is $c:b<c<a$ ) and we can construct a sequence of points converging to $a$ . This is not a trivial fact, I think (and I don’t remember it was covered anywhere before). Indeed, if we consider $\overline{S}_\Omega$ , for example, then $\Omega$ is a limit point of $S_\Omega$ but there is no sequence converging to it (any sequence is a countable subset and has an upper bound in $S_\Omega$ ). However, in the case of $S_\Omega$ we prove it as follows: $S_a$ is countable; let us enumerate all its elements with positive integer indexes: $S_a=\{s_n\}_{n=0,1, . . .}$ ; then let $n_0$ be such that $s_{n_0}=a_0$ , let $n_1$ be the minimum index greater than $n_0$ such that $s_{n_1}>s_{n_0}$ , let $n_2$ be the minimum index such that $n_2>n_1$ and $s_{n_2}>s_{n_1}$ , etc. (we can always do this as there is no largest element in $S_a$ , therefore, there is infinite number of elements greater than any $x\in S_a$ ); note that for every $k$ , $n_k\ge k$ and $s_{n_k}\ge s_m$ for all $m\le n_k$ ; if $x\in S_a$ then $x=s_k$ for some $k$ and $s_{n_m}\ge s_{n_k}\ge s_k=x$ for all $m\ge k$ . Now, once we have the sequence converging to $a$ , by using (b), $[a_0\times 0,a\times 0)$ has the order type of $[0,1)$ . By the principle of transfinite induction, this holds for any $a>a_o$ .(d) We show that any point is path connected to $(a_1,0)$ where $a_1$ is the immediate successor of $a_0$ . Indeed, $(a_0,x)$ for $x>0$ is path connected to $(a_1,0)$ ($((a_0,0),(a_1,0)]$ is homeomorphic to $(0,1]$ ) and for $b>a_0$ any point $(b,x)$ , $x\ge 0$ , is path connected to $(b,0)$ which is path connected to $(a_1,0)$ (by (a) and (c), $[(a_1,0),(b,0))$ has the same order type as $[0,1)$ and the order topology on a convex subset is the same as the subspace topology). Therefore, $L$ is path connected as the union of path connected subsets having a common point.(e) An open interval $(-\infty,(b,0))$ in $L$ has the order type of an open interval in $\mathbb{R}$ . Indeed, by (c), $[(a_0,0),(b,0))$ has the order type of $[0,1)$ . Since for convex subsets the order topology is the same as the subspace topology, $((a_0,0),(b,0))$ has the order type of $(0,1)\subseteq\mathbb{R}$ . Therefore, for any $(a,x)$ there is an open neighborhood $(-\infty,(a+1,0))$ with the order type of $(0,1)$ . Moreover, there is a bijective order preserving function from one set onto the other, and by 7(a), it is a homeomorphism.(f) If $L$ could be embedded into $\mathbb{R}^n$ then $L$ would be homeomorphic to a subspace of $\mathbb{R}^n$ which has a countable basis for the topology. Therefore, there must be a countable basis of the topology of $L$ . We show that there is no. For any $a$ : $((a,0),(a+1,0))$ is open and contains $(a,1/2)$ but does not contains $(b,1/2)$ for $b\neq a$ . Therefore, there is a basis element such that it contains $(a,1/2)$ but does not contain $(b,1/2)$ for $b\neq a$ . Call it $B_a$ . Note that for $a\neq b$ : $B_a\neq B_b$ . Since $S_\omega$ is uncountable, there must be uncountable number of distinct sets in the basis.