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 is an element of A subset of S, then the base of neighborhoods of the point x is a family of intervals {[a, b) : a, b is an element of R, a < b (sic) x is an element of[a, b)}. If x is an element of S \ A, then the base of neighborhoods of x is {(c, d]: c, d is an element of R, c < d (sic) x is an element of(c, d]}. It is proved that for a countable subset A subset of 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 of 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 R. Then the space S-A is homeomorphic to the space S. Let a countable set A subset of R be such that its closure (A) over bar 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.