On some linearly ordered topological spaces homeomorphic to the Sorgenfrey line

E. S. Sukhacheva ; T. E. Khmyleva

Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 5 (2014), pp. 63-68

Résumé

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 $S$.

Détail

Keywords: Sorgenfrey line, derivative set, homeomorphism, ordinal.