Some Types of Mappings in Bitopological Spaces

This work, introduces some concepts in bitopological spaces, which are nm-j-ω-converges to a subset, nm-j-ω-directed toward a set, nm-j-ω-closed mappings, nm-j-ω-rigid set, and nm-j-ω-continuous mappings. The mainline idea in this paper is nm-j-ω-perfect mappings in bitopological spaces such that n = 1,2 and m =1,2 n ≠ m. Characterizations concerning these concepts and several theorems are studied, where j =  , δ,  , pre, b, .


Introduction and Preliminaries:
In 1963 Kelly J. C. (1) introduced the definition, a set G with two topologies σ 1 and σ 2 is said to be bitopological space and denoted by (G, σ 1 , σ 2 ) and a subset K ⊆ G. The closure and interior of K in (G, σ n ) is denoted by σ n -cl(K) and σ n -int(K), where n = 1, 2. A topological space (G, σ) and a point g in G is said to be condensation point of K ⊆ G if every open neighborhood S in σ with g  S, the set K ∩ S is uncountable )2). In 1982 the ω-closed set was first exhibited by H. Z. Hdeib in (3) defined it as a subset K ⊆ G is called ω-closed if it incorporates each its condensation points, and the ω-open set is the complement of the ω-closed set and the ω-closed of the set K ⊆ G denoted by cl (K). The ω-interior of the set K ⊆ G is defined as the union of all ω-open sets content in K and is denoted by int (K). In (4) a point g  G is said to θ-cluster points of K ⊆ G if cl(S) ∩ K ≠ φ for each open set S of G contained g.Al so in (4) the set of each θ-cluster points of K is called the θ-closure of K and is denoted by cl (K). A subset K ⊆ G is called θ-closed (4) if K = cl (K). The complement of θ-closed set is said to be θ-open. A point g  G is said to θ-ω-cluster points of K ⊆ G if cl (S) ∩ K ≠ φ for each ω-open set S of G containing g. The set of each θ-ω-cluster points of K is called the θ-ωclosure of K and is denoted by cl ( ). A subset K ⊆ G is called θ-ω-closed (4) if K = cl ( ). The complement of θ-ω-closed set is said to be θ-ωopen. A subset K ⊆ G is said to be δ-closed (5) if K = cl (K) = {g  G : int(cl(S)) ∩ K ≠ φ, S  τ and g  S}. The complement of δ-closed is called δ-open set, and K is δ-ω-closed if K = cl ( ) = {g  G : int (cl(S)) ∩ K ≠ φ, S  τ and g  S}. For other notions or notations not defined here, R. Englking (6) should be followed closely. Several characterizations of ω-closed sets were provided in (4, 5, 8, 9, and 10). Some of the results in (11), (12), (13), (14) and (15) will be bult.
The filter generated by a filter base  consists of all supersets of elements of . An open filter base on a space G is a filter base with open members.
The set  g of all neighborhoods (nbds) of g Theorem 1. In a bitopological space (G, σ 1 , σ 2 ) a point g is an nm-j-ω-condensation of a filter base  on G if there subsistent a filter base * finer than  such that *nmj -ω g, where j =  , δ,  , pre, b, .
Proof: () Let g be an nm-j-ω-condensation point of a filter base  on G, then every σ n -open nbd S of g, the j-ω-closure of S contains a member of  and thus contains a member of any filter base * minutes than , so that * nmj -ω g.
() Assume that g is not an nm-j-ω-condensation point of a filter base  on G, then there subsistent an σ n -open nbd S of g, such that j-ω-closure of S contains no member of , denote by * the family of sets M* = M ∩ (G -( cl (S)) for M  , then the sets M* are nonempty. And * is a filter base and indeed it is minute than , since M 1 * = M 1 ∩ (G  cl (S)) and M 2 * = M 2 ∩ (G  cl (S)), so there is an M 3  M 1 ∩ M 2 and this lead to: By construction * not nm-j-ω-convergent to g. This contradiction, and thus g is an nm-j-ωcondensation point of a filter base  on G.

Definition 7.
A filter base  on a bitopological space (G, σ 1 , σ 2 ) is said to be nm-j-ω-directed toward to a set K  G (written as nmj-ω -dir-tow  K) if for each filter base  finer  has an nm-j-

Theorem 2.
Let  be a filter base on a bitopological space (G, σ 1 , σ 2 ) and point g  G , : M } is a filter base on G finer than , and conspicuously g  nm-j-ω-cod. So  cannot be nm-j-ω-directed towards g.
is said to be nm-j-ω-perfect if for every filter base  on (G), nm-j-ω-directed towards some subset L of (G), the filter base Then  is a filter base on G finer than  −1 (). Since  nmj -ω-dir-tow  h, and by . Thus,  is nm-j-ω-perfect.
, for n, m= 1and 2 such that (n  m), and for Proof: Straightforward.
there is S  σ n and M  , such that K  S and cl (S) ∩ M = . or equivalent, if for every filter base  on G whenever, The relation between weakly and strongly nm-j-ωcontinuous mappings are given by the following The identity mapping  : (G, τ r )  (G, τ hdis ) . Then,  is weakly nm-j-ω-continuous mapping but it is not nm-j-ω-continuous mapping.   is super nm-j-ω-continuous mapping but it is not strongly nm-j-ω-continuous mapping.

(b) To prove  −1 (h) is nm-j-ω-rigid, let h  H, and
assume that  be a filter base on G such that (nm-

Study on some Types of j-ω-perfect Mappings in Bitopological Spaces.
In this section, nm-j-ω-perfect mappings are given and used the definitions of characterizations theorems for an nm-j-ωcontinuous mapping and weakly nm-j-ω-continuous mapping and strongly nm-j-ω-continuous mapping and super nm-j-ω-continuous mapping and almost nm-j-ω-continuous mapping are indicated to this end, and n ,m = 1, 2 where j =  , δ,  , pre, b, .
By analogy to Theorem (20), amplest condition for a mapping to be almost nm-j-ω-perfect, is prove as follows.

Conclusion.
The main purpose of the present work is the starting point for some application of pairwise supra-ω-perfect mappings of abstract topological structures in filter base by using bitopological spaces. Definitions of characterizations theorems are used for an nm-j-ω-continuous mapping and weakly nm-j-ω-continuous mapping and strongly nm-j-ω-continuous mapping and super nm-j-ωcontinuous mapping and almost nm-j-ω-continuous mapping.