On some linearly ordered topological spaces homeomorphic to the Sorgenfrey line

In this paper, we consider a topological space $S_A$ which is a modification of the Sorgenfrey line $S$ and is defined as follows: if a point $x\in A\subset S$, then the base of neighborhoods of the point $x$ is a family of intervals $\{[a,b)\colon a,b\in\mathbb R,\ a<b,\ \text{and}\ x\in[a,b)\}$. If $x\in S\setminus A$, then the base of neighborhoods of $x$ is $\{(c,d]\colon c,d\in\mathbb R,\ c<d\ \text{and}\ x\in(c,d]\}$. It is proved that for a countable subset $A\subset\mathbb R$ the closure of which in the Euclidean topology is a countable space, the space $S_A$ is homeomorphic to the space $S$. In addition, it was found that the space $S_A$ is homeomorphic to the space$S$ for any closed subset $A\subset\mathbb R$. Similar problems were considered by V. A. Chatyrko and Y. Hattori in [4], where the “arrow” topology on the set $A$ was replaced by the Euclidean topology. In this paper, we consider two special cases: $A$ is a closed subset of the line in the Euclidean topology and the closure of the set $A$ in the Euclidean topology of the line is countable.
The following results were obtained:
Let a set $A$ be closed in $\mathbb R$. Then the space $S_A$ is homeomorphic to the space $S$.
Let a countable set $A\subset\mathbb R$ be such that its closure $\overline A$ is countable relatively to $\mathbb R$. Then $S_A$ is homeomorphic to $S$.
Let $A$ be a countable closed subset in $S$. Then $S_A$ is homeomorphic to $S$.