On the homeomorphism of the Sorgenfrey line and its modifications $S_{\mathcal{Q}}$

In this paper, it is proved that two topological spaces, namely, the Sorgenfrey line $S$ and its modifications $S_{\mathcal{Q}}$, where $\mathcal{Q}$ is the set of rational numbers on the real line, are nonhomeomorphic. Topology of the space $S_{\mathcal{Q}}$ is defined as follows: if $x\in\mathcal{Q}\subset S$, then the base of neighborhoods of the point $x$ is the family of semiintervals $\{[x, x+\varepsilon):\varepsilon>0\}$, and if $x\in S\setminus\mathcal{Q}$, then the base of the neighborhood is a family of semiintervals $\{(x-\varepsilon, x]:\varepsilon>0\}$. The proof of this fact uses monotonicity of the homeomorphism $\varphi: S\to S$ on some interval $(a, b)\subset S$ (E. K. Van Douwen, 1979).