Abstract:
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.
Keywords:Sorgenfrey line, point of condensation, Baire space, nowhere dense set, homeomorphism, ordinal, spaces of the first and second category, $F_\sigma$-set, $G_\delta$-set.
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