Geodesic
On some linearly ordered topological spaces homeomorphic to the Sorgenfrey line
Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 5 (2014), pp. 63-68 Cet article a éte moissonné depuis la source Math-Net.Ru

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In this paper, we consider a topological space $S_A$ which is a modification of the Sorgenfrey line $S$ and is defined as follows: if a point $x\in A\subset S$, then the base of neighborhoods of the point $x$ is a family of intervals $\{[a,b)\colon a,b\in\mathbb R,\ a. If $x\in S\setminus A$, then the base of neighborhoods of $x$ is $\{(c,d]\colon c,d\in\mathbb R,\ c. It is proved that for a countable subset $A\subset\mathbb R$ the closure of which in the Euclidean topology is a countable space, the space $S_A$ is homeomorphic to the space $S$. In addition, it was found that the space $S_A$ is homeomorphic to the space$S$ for any closed subset $A\subset\mathbb R$. Similar problems were considered by V. A. Chatyrko and Y. Hattori in [4], where the “arrow” topology on the set $A$ was replaced by the Euclidean topology. In this paper, we consider two special cases: $A$ is a closed subset of the line in the Euclidean topology and the closure of the set $A$ in the Euclidean topology of the line is countable. The following results were obtained: Let a set $A$ be closed in $\mathbb R$. Then the space $S_A$ is homeomorphic to the space $S$. Let a countable set $A\subset\mathbb R$ be such that its closure $\overline A$ is countable relatively to $\mathbb R$. Then $S_A$ is homeomorphic to $S$. Let $A$ be a countable closed subset in $S$. Then $S_A$ is homeomorphic to $S$.
Keywords: Sorgenfrey line, derivative set, homeomorphism, ordinal. 
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     title = {On some linearly ordered topological spaces homeomorphic to the {Sorgenfrey} line},
     journal = {Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika},
     pages = {63--68},
     year = {2014},
     number = {5},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VTGU_2014_5_a5/}
}
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E. S. Sukhacheva; T. E. Khmyleva. On some linearly ordered topological spaces homeomorphic to the Sorgenfrey line. Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 5 (2014), pp. 63-68. http://geodesic.mathdoc.fr/item/VTGU_2014_5_a5/

[1] Aleksandrov P. S., Vvedenie v teoriyu mnozhestv i obshchuyu topologiyu, Nauka Publ., Moscow, 1977, 368 pp. (in Russian) | MR

[2] Kuratovskiy K., Mostovskiy A., Teoriya mnozhestv, Mir Publ., Moscow, 1970, 416 pp. (in Russian) | MR

[3] Engel'king R., Obshchaya topologiya, Mir Publ., Moscow, 1986, 752 pp. (in Russian) | MR

[4] Chatyrko V. A., Hattori Y., “A poset of topologies on the set of real numbers”, Comment. Math. Univ. Carolin., 54:2 (2013), 189–196 | MR | Zbl