ON PRE GENERALIZED b-CLOSED MAP IN TOPOLOGICAL SPACES

In this paper, we introduce a new class of pre generalized b-closed map in topological spaces (briefly pgb-closed map) and study some of their properties as well as inter relationship with other closed maps. AMS Subject Classification: 54C05, 54C08, 54C10


Introduction
Different types of Closed and open mappings were studied by various researchers. In 1996, Andrijevic introduced new type of set called b-open set. A.A.Omari and M.S.M. Noorani [1] introduced and studied b-closed map.
The aim of this paper is to introduce pre generalized b-closed map and to continue the study of its relationship with various generalized closed maps. Through out this paper (X, τ ) and (Y, σ) represent the non-empty topological spaces on which no separation axioms are assumed, unless otherwise mentioned.

Preliminaries
In this section, we referred some of the closed set definitions which was already defined by various authors. Definition 2.5. [6] Let a subset A of a topological space (X, τ ), is called a generalized closed set (briefly g-closed) if cl(A) ⊆ U whenever A ⊆ U and U is open in X.
Definition 2.8. [9] Let a subset A of a topological space (X, τ ), is called a generalized α*-closed set (briefly gα *-closed) if αcl(A) ⊆ intU whenever A ⊆ U and U is α open in X.

On Pre Generalized b-Closed Map
In this section, we introduce pre generalized b -closed map (pgb -closed map) in topological spaces by using the notions of pgb -closed sets and study some of their properties.  Proof. Let f : (X, τ ) → (Y, δ) is closed map and V be an closed set in X then f (V ) is closed in Y . Hence pgb -closed in Y . Then f is pgb -closed.
The converse of above theorem need not be true as seen from the following example.
The converse of above theorem need not be true as seen from the following example.
Theorem 3.6. Every gα -closed map is pgb -closed but not conversely.
Proof. Let f : (X, τ ) → (Y, σ) be gα-closed map and V be an closed set in The converse of above theorem need not be true as seen from the following example.
The converse of above theorem need not be true as seen from the following example.
Theorem 3.10. Every g -closed map is pgb -closed but not conversely.
Proof. Let f : (X, τ ) → (Y, δ) g -closed map and V be an closed set in X The converse of above theorem need not be true as seen from the following example.
Theorem 3.12. Every gαb -closed map is pgb -closed but not conversely.
Proof. Let f : (X, τ ) → (Y, σ) be gαb -closed map and V be an closed set in X then f (V ) is closed in Y . Hence pgb -closed in Y . Then f is pgbclosed.
The converse of above theorem need not be true as seen from the following example. Theorem 3.14. Every rgb -closed map is pgb -closed but not conversely.
Proof. Let f : (X, τ ) → (Y, δ) be rgb closed map and V be an closed set in The converse of above theorem need not be true as seen from the following example.
Theorem 3.16. Every pgb -closed map is pgb -closed but not conversely.
Proof. Let f : (X, τ ) → (Y, δ) be pgb -closed map and V be an closed set in X then f (V ) is closed in Y . Hence gb -closed in Y . Then f is gb -closed.
The converse of above theorem need not be true as seen from the following example.  Proof. Let f : (X, τ ) → (Y, δ) be sg -closed map and V be an closed set in The converse of above theorem need not be true as seen from the following example.   Proof. Let F be a closed set of A then F is pgb -closed set. By theorem Here f A is pgb -closed and also continuous.
Proof. Let F be any closed set in (X, τ ). Since f is closed map, f (F ) is closed set in (Y, σ). Since g is pgb -closed map, g(f (F )) is pgb -closed set in (Z, η). That is g · f (F ) = g(f (F )) is pgb closed. Hence g · f is pgb closed map.
Proof. Suppose f is pgb -closed. Let S ⊂ Y and U be an open set of (X, τ ) Conversely, Let F be a closed set of (X, τ ). Then f −1 (f (F c )) ⊂ F c and F c is an open in (X, τ ). By hypothesis, there exist a pgb -open set V in (Y, σ) such Theorem 3.26. Let h : X → X 1 × X 2 be pgb -closed map and Let (ii)⇒(iii) Let F be a closed set of (X, τ ). Then F c is open set in (X, τ ).

On Pre Generalized b-Open Map
In this section, we introduce pre   The converse of above theorem need not be true as seen from the following example.  Proof. Suppose f is pgb -closed set. Let S ⊂ Y and U be an open set of (X, τ ) such that Conversely, Let F be a closed set of (X, τ ). Then f −1 (f (F c )) ⊂ F c and F c is an open in (X, τ ). By hypothesis, there exists a pgb open set V in (Y, σ) such σ) and there fore f is pgb -closed. (ii)⇒ (iii) Let F be closed set of (X, τ ), Then F c is open set in (X, τ ). By assumption, f (F c ) is pgb -open in (Y, σ). There fore f (F ) is pgb -closed in (Y, σ). Hence f is pgb-closed.