О некоторых линейно упорядоченных топологических пространствах, гомеоморфных прямой Зоргенфрея

Вестник Томского государственного университета. Математика и механика. 2014. № 5 (31). С. 63–68.

Хмылева Т. Е., Сухачева Е.С.

Sukhacheva E.S., Khmyleva T.E. ON SOME LINEARLY ORDERED TOPOLOGICAL SPACES
HOMEOMORPHIC TO THE SORGENFREY LINE
In this paper, we consider a topological space SA which is a modification of the Sorgenfrey
line S and is defined as follows: if a point x∈ A ⊂ S , then the base of neighborhoods of the point
x is a family of intervals {[a,b) : a,b∈􀁜,a < bи x∈[a,b)} . If x∈S \ A , then the base of
neighborhoods of x is {(c,d ]: c,d ∈􀁜,c < d и x∈(c,d ]} . It is proved that for a countable subset
A ⊂ 􀁜 the closure of which in the Euclidean topology is a countable space, the space SA is
homeomorphic to the space S. In addition, it was found that the space SA is homeomorphic to the
space S for any closed subset A ⊂ 􀁜 . 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 􀁜 . Then the space SA is homeomorphic to the space S.
Let a countable set A ⊂ 􀁜 be such that its closure A is countable relatively to 􀁜 . Then SA
is homeomorphic to S .
Let A be a countable closed subset in S. Then SA is homeomorphic to S .

Keywords: топология, Topology