On some linearly ordered topological spaces homeomorphic to the sorgenfrey line

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 e A с S , then the base of neighborhoods of the point x is a family of intervals {[a, b): a, b e R, a < b и x e[a,b)}. If x e S \ A , then the base of neighborhoods of x is {(c,d]: c,d e R,c < dиx e (c,d]} . Itis proved that for a countable subset A с К. 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 с К.. 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 R. Then the space S _A is homeomorphic to the space S. Let a countable set A с К. be such that its closure A is countable relatively to R. Then S _A is homeomorphic to S . Let A be a countable closed subset in S. Then S _A is homeomorphic to S .

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