TAILIEUCHUNG - Countable dense homogeneous bitopological spaces
In this paper we shall introduce the concept of being countable dense homogeneous bitopological spaces and define several kinds of this concept. We shall give some results concerning these bitopological spaces and their relations. | Tr. J. of Mathematics 23 (1999) , 233 – 242. ¨ ITAK ˙ c TUB COUNTABLE DENSE HOMOGENEOUS BITOPOLOGICAL SPACES Abdalla Tallafha, Adnan Al-Bsoul, Ali Fora Abstract In this paper we shall introduce the concept of being countable dense homogeneous bitopological spaces and define several kinds of this concept. We shall give some results concerning these bitopological spaces and their relations. Also, we shall prove that all of these bitopological spaces satisfying the axioms p-T 0 and p-T 1 . AMS 1991 classification: 54E55, 54D10, 54G20. Key words: and phrases: CDH, bitopological spaces. 1. Introduction Countable dense homogeneous spaces were introduced by Bennett [1]. Recall that a topological space (X, τ ) is called countable dense homogeneous (CDH) iff X is separable and, if D1 and D2 are two countable dense subsets of X, then there is a homeomorphism h: X→ X such that h(D1 )=D2 . In 1963, Kelly [4] introduced the concept of bitopological spaces. A set X equipped with two topologies is called a bitopological space. Let X be any set. By τcof , τdis , τind and τu , , . (in the case X=IR), we mean the cofinite, the discrete, the indiscrete, the usual Euclidean, the left ray, and the right ray topologies, respectively. Let (X, τ ) be a topological space, A⊆X. By τA we mean the relative topology on A. If (X, τ1 , τ2 ) is a bitopological space and A⊆X, cl i (A) will denote the closure of A with respect to τi ; i= 1, 2. A subset D in (X, τ1 , τ2 ) is called 233 TALLAFHA, AL-BSOUL, FORA dense if cl1 (D)=cl2 (D)=X. A bitopological space (X, τ1 , τ2 ) is called separable if both topological spaces (X, τ1 ) and (X, τ2 ) are separable. For a set A, we shall denote the cardinality of A by
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