On Generalised Pre Regular Weakly(gprw)-closed and Gprw-quasi Closed Functions in Topological Spaces

The aim of this paper is to introduce and study gprw-closed and gprw-quasi closed functions in topological spaces. Also some properties of these classes of functions will be discussed here.


Introduction
In 1970, Levine [6] initiated the study of so called generalized closed sets.This notion has been studied extensively in the recent years by many topologists because generalized closed sets are not only natural generalization of closed sets.More over, they also suggest several new properties of topological spaces.Most of these new properties are separation axioms weaker than T 1 , some of them found to be useful in computer science and digital topology.Furthermore, the study of generalized closed sets also provides new characterization of some known classes of spaces.In 1997 Y. Gnanambal [4] proposed the definition of generalized pre regular-closed sets (briefly gpr-closed) and further notion of pre regular T 1/2 space and generalized pre regular continuity was introduced.And in 2007, notion of regular weakly closed set is defined by S.S. Benchalli and R.S. Wali [1] and they proved that this class lie between the class of all w-closed sets [11] and the class of all regular generalized closed sets [10] and some of its properties are investigated in it.Now in this paper, the notion of gprw-closed functions and gprw-quasi closed functions are defined by using the notions of gprw-closed sets which is introduced by S. Mishra, V. Joshi and N. Bhardwaj [8].Also properties of these classes of functions are studied in this paper.[9] set if A = int(cl(A)) and regular closed (briefly r-closed) [9]

Preliminary Notes Definition 2.1 A subset A of X is called regular open (briefly r-open)
and pre-closed [7] The family of all regular semiopen sets of X is denoted by RSO(X).Definition 2.4 Let (X, τ ) be a topological space and A be a subset of X said to be gprw-closed [8] if pcl(A) ⊂ U, whenever A ⊂ U and U is regular semi open.The family of all gprw-closed set in space X is denoted by GP RW C(X).Definition 2.5 A subset of a topological space (X, τ ) is called 1. Generalized pre regular closed (briefly gpr-closed) [4] if pcl(A) ⊆ U whenever A ⊆ U and U is regular open in X. [11] if cl(A) ⊆ U whenever A ⊆ U and U is semiopen in X.

Weakly closed (briefly w-closed)
4. Regular weakly closed (briefly rw-closed) [1] if cl(A) ⊆ U whenever A ⊆ U and U is regular semiopen in X.We denote the set of all rw-closed sets in X by RW C(X).
Theorem 2.6 Every regular semiopen set in X is semiopen but not conversely.
Theorem 2.7 [5] If A is regular semiopen in X, then X \ A is also regular semiopen.

gprw-closed and gprw-quasi closed Functions in Topological Spaces
Proof.Let H be a closed set in (X, τ ).Then f (H) is closed in (Y, σ), by the definition of closed map.Now gof (H) = g(f (H)) and by the definition of g we have g(f (H)) = gof (H) is gprw-closed (Z, η).Thus from an arbitrary closed set H in (X, τ ) and we have its image under gof is gprw-closed in (Z, η).Proof.Let f : (X, τ ) → (Y, σ) be quasi gprw-closed function on two topological spaces.Now let us consider any closed set F of (X, τ ), By [8] And to show that it is gprw-closed we consider a closed set V and then it is gprw-closed by [8], now its image under quasi gprw-closed function f is closed in (Y, σ), and it will be gprw-closed in Y .Therefore f is gprw-closed.

Example 3.8 Let a topological spaces
Then this function is gprw-closed but not gprw-quasi closed, since the image of gprwclosed set is not closed for example {1} has image {2} and {2, 3} has image {1, 3} which are not closed.

Theorem 3.9 A function f : X → Y said to be quasi gprw-closed if and only if for every subset U, we have cl(U) ⊂ f (gprw − cl(U)).
Proof.As U ⊂ gprw − cl(U), for an arbitrary set U, therefore by taking f on both sides we get [8], it is gprw-closed further g : Y → Z is also gprw-closed function, so g(f (B)) is gprw-closed set in topological space Z. Hence we consider an arbitrary gprw-closed set in X and got its image closed under gof therefore gof : X → Z is quasi gprw-closed function.

Definition 3.13 A function
Theorem 3.14 Let f : X → Y and g : Y → Z be two functions on topological spaces then Proof.
1. Let closed subset F of topological space X, then by the given fact that Hence image of an arbitrary closed set in X under gof is closed in Z, so gof : X → Z is closed function.Proof.
This implies g is closed function.
2. Suppose that F be any arbitrary gprw-closed set in X.Since gof is quasi gprw-closed therefore gof (F ) is closed set in Z.And also g is gprwcontinuous injective function therefore and hence closed in Y .Corollary 3.17 Let X and Y be topological space.Then a surjective function g :

Corollary 3.18 Let g : X → Y be a gprw-continuous, quasi gprw-closed and surjective function then the topology on
Proof.As gprw − int(U) ⊂ U, for an arbitrary set U, then we have

For each subset
3. For each x ∈ X and each gprw-neighbourhood U of x ∈ X, there exist a neighbourhood V of f (x) in Y such that V ⊂ f (U).
(2) ⇒ (3) Let x ∈ X and U be an arbitrary gprw-neighbourhood of x in X.
Then there exist a gprw-open set V in X such that x ∈ V ⊂ U. Then by (2) we have f (V ) = f (gprw − int(V )) ⊂ int(f (V )) and as for an arbitrary set we have int(f (V )) ⊂ f (V ), therefore we arrive at f (3) ⇒ (1) Let U be an arbitrary gprw-open set X. Now for each y ⊂ f (U) by ( 3) there exist neighbourhood

Theorem 3 . 2 A
function f : (X, τ ) → (Y, σ) is said to be gprw-closed if and only if for each subset S of Y and for each open set

Theorem 3 . 6
Every quasi gprw-closed function is closed as well as gprwclosed function.

Theorem 3 .
22 For a function f : X → Y the following statements are equivalent 1. f is quasi gprw-open.
let U be a gprw-open set of X and put B = Y \ f (U).Then X \ U is a gprw-closed set in Y containing f −1 (B).By hypothesis there exist a closed set F of Y such that B ⊂ F and f −1 (F ) ⊂ X \ U. Hence we obtain f (U) ⊂ Y \F .On the other hand, it follows that B ⊂ F , Y \F ⊂ Y \B = f (U).Thus we got f (U) = Y \ F which is open and hence f is quasi gprw-open function.Theorem 3.24 A functionf : X → Y is quasi gprw-open if and only if f −1 (cl(B)) ⊂ gprw − cl(f −1 (B)) for every subset B of Y .Proof.For any subset B of Y , f −1 (B) ⊂ gprw − cl(f −1 (B)).Therefore by the above theorem there exists a closed set F in Y such that B ⊂ F andf −1 (B) ⊂ gprw − cl(f −1 (B)).Therefore we obtained f −1 (cl(B)) ⊂ f −1 (F ) ⊂ gprw − cl(f −1 (B)).Conversely let B ⊂ Y and F be a gprw-closed set of X containing (f −1 (B)).Put W = cl Y (B), then we have B ⊂ W and W is closed set and f −1 (W ) ⊂ gprw − cl(f −1 (B)) ⊂ F .Then by theorem above f is quasi gprw-open.Theorem 3.25 Let f : X → Y and g : Y → Z be two functions such that gof : X → Z is quasi gprw-open function and g is continuous injective then f is quasi gprw-open.Proof.For gprw-open set U in X, we have gof (U) is open in Z. Since gof is quasi gprw-open and g is an injective continuous function, f (U) = g −1 (gof (U)) is open in Y .Hence f is quasi gprw-open.
2. Let us consider gprw-closed set F of topological space X then by definition of f which is gprw-closed we have f (F ) is closed set in Y .Since g is gprw-closed function therefore g(f (F )) = gof (F ) is gprw-closed set in Z. Hence gof : X → Z is gprw * * -closed function.3. Consider any gprw-closed set F in X then by definition of f which is gprw * * -closed function, the set f (F ) is gprw-closed in Y .Now we are given g : Y → Z is quasi gprw-closed set, therefore g(f (F )) = gof (F ) is closed set of Z. Now if we take a gprw-closed in X then its image under gof is closed set in Z. Hence gof : X → Z is quasi gprw-closed function.Let f : X → Y and g : Y → Z be two functions such that gof : X → Z is quasi gprw-closed function on topological spaces then 1.If f is gprw-irresolute and surjective function then g is closed function.2. If g is gprw-continuous and injective function, then f is gprw * * -closed function.
which is open set in Y .This shows that f is quasi gprw-open function.
Theorem 3.23 A function f : X → Y is quasi gprw-open if and only if for any subset B of Y and for any gprw