On modification of the Sorgenfrey line

In this paper, we consider a topological space $S_A$ that is a modification of the Sorgenfrey line $S$ and is defined as follows: if a point $x\in A\subset \mathbf{R}$, then the base of neighborhoods of the point is $\{[x, x+\varepsilon), \forall\varepsilon>0\}$; if a point $x\in \mathbf{R}\setminus A$, then the base of neighborhoods of the point is $\{(x-\varepsilon, x], \forall\varepsilon>0\}$. The following criterion for a homeomorphism of the spaces $S_A$ and $S_Q$ has been obtained: the spaces $S_A$ and $S_Q$ are homeomorphic if and only if a subset $A\subset S_A$ is countable and dense in $S$.