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

E. S. Sukhacheva ; T. E. Khmyleva

Matematičeskie zametki, Tome 103 (2018) no. 2, pp. 258-272

Résumé

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.

Détail

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