On a Homeomorphism between the Sorgenfrey Line $S$ and Its Modification $S_P$

A topological space $S_P$, which is a modification of the Sorgenfrey line $S$, is considered. It is defined as follows: if $x\in P\subset S$, then a base of neighborhoods of $x$ is the family $\{[x,x+\varepsilon),\,\varepsilon>0\}$ of half-open intervals, and if $x\in S\setminus P$, then a base of neighborhoods of $x$ is the family $\{(x-\varepsilon,x],\,\varepsilon>0\}$. A necessary and sufficient condition under which the space $S_P$ is homeomorphic to $S$ is obtained. Similar questions were considered by V. A. Chatyrko and I. Hattori, who defined the neighborhoods of $x \in P$ to be the same as in the natural topology of the real line.