On Pgprw-closed maps and pgprw-open maps in topological spaces

In this paper, we introduce pgprw-closed map from a topological space X to a topological space Y as the image of every closed set is pgprw-closed and also we prove that the composition of two pgprw-closed maps need not be pgprw-closed map. We also obtain some properties of pgprw-closed maps.

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Introduction

Generalized closed mappings were introduce and studied by Malghan [12] . Rg-closed maps by Arockiaran [13] . In this paper, a new class of maps called pre generalized pre regular weakly closed maps (briefly, pgprw-closed) maps have been introduced and studied their relations with various generalized closed maps. We prove that the composition of two pgprw -closed maps need not be pgprw-closed map. We also obtain some properties of pgprw-closed maps. Wali and Vivekananda Dembre [14] introduced new class of sets called pre generalized pre regular weakly -closed (briefly pgprw -closed) sets in topological spaces which lies between the class of all p -closed sets and the class of all gpr -closed sets

Preliminaries

Throughout this paper space (X, τ) and (Y, σ) (or simply X and Y) always denote topological space on which no separation axioms are assumed unless explicitly stated. For a subset A of a space X, Cl(A), Int(A), A c , P-Cl(A) and P-int(A) denote the Closure of A, Interior of A, Compliment of A, pre-closure of A and pre-interior of (A) in X respectively.

Definition:

A subset A of a topological space (X,τ) is called 1. Semi-open set [1] if A  cl(int(A)) and semi-closed set if int(cl(A))  A..

2.

Semi-pre open set [2] (= β-open [1] if Acl(int(cl(A)))and a semi-pre closed set(=β-closed) if int(cl(int(A)))A. 3. Pre-open set [3] if A  int(cl(A)) and pre-closed set if cl(int(A))  A. 4. Generalized pre closed(briefly gp-closed)set [4] ifpcl(A)⊆U wheneverA⊆U and U is open in X. 5. Generalized pre regular closed set(briefly gpr-closed) [5] if pcl(A)U whenever AU and

U is regular open in X. 6. Generalized semi pre regular closed (briefly gspr-closed) set [6] if spcl(A)⊆U whenever A⊆U and U is regular open in X. 7. Pre-generalized-pre-regular closed(briefly pgpr-closed)set [7] if pcl(A)⊆U whenever A⊆ U and U is rg-open in X. 8. W-closed set [8] if cl(A)  U whenever A U and U is semi-open in X. 9. Regular generalized closed set(briefly rg-closed) [9] if cl(A)U whenever AU and U is regular open in X. 10. Generalized closed set(briefly g-closed) [10] if cl(A)  U whenever A  U and U is open in X. 11. Pre generalized pre regular ωeakly closed set [14](briefly pgprω-closed set) if pCl(A)U whenever AU and U is rgα open in (X, τ). 12. pre generalized pre-regular ωeakly open(briefly pgprω-open) [15] set in X if A c is pgprωclosed in X.

On Pgprw-Closed Maps

On Pgprw-Closed Maps

Definition

) is said to be Pre generalized pre regular weakly closed map (briefly pgprw-closed) if the image of every closed set (X, τ) is pgprw-closed in (Y, σ).

Theorem:

Every closed map is pgprw-closed map but not conversely.

Proof:

The proof follows from the definitions and fact that every closed set is pgprw-closed set from Theorem 2.4 [14] .

Remark:

The converse of the above theorem need not be true as seen from the following example.

Example:

Consider X=Y= {a,b,c} with topologies τ = {X, ϕ,{a,c}} and σ = { Y, ϕ, {a}}. Let f: (X, τ) (Y, σ) be the identity map. Then this function is pgprw-closed but not closed as the image of closed set {b} in X is {b} which is not closed set in Y.

Theorem: Every p-closed map is pgprw-closed map but not conversely.

Proof: The proof follows from the definitions and fact that every p-closed set is pgprw-closed set from Theorem 2.4 [14] .

Remark:

The converse of the above theorem need not be true as seen from the following example.

Example:

Consider X=Y={a,b,c,d} with topologies τ = {X, ϕ,{a},{b},{a,b},{a,b,c}} and σ ={Y, ϕ,{a},{b},{a,b},{a,b,c}}. Let map f: (X, τ) (Y, σ) defined by f(a)=c,f(b)=c,f(c)=b,f(d)=d. Then f is pgprw closed map but not closed map and not p-closed map as the image of closed set F={c,d}in X then f(F)={b,d} in Y which is not p-closed set in Y.

Theorem:

If a map f: (X, τ) (Y, σ) is closed then the following holds. If f is pgprw closed map then f is gp, gpr, gspr closed map but not conversely.

Proof:

The proof follows from the defintions and fact that every pgprw closed set is gp, gpr, gspr closed set from Theorem 2.4 [14] .

Remark:

The converse of the above theorem need not be true as seen from the following example.

Example:

Consider X=Y={a,b,c} with topologies τ = {X, ϕ,{a},{b,c}} and σ ={Y, ϕ,{a}.Let map f: X, τ) (Y, σ) defined by f (a)=b, f (b)=a, f (c)=c. Then f is gp, gpr, gspr, closed map but not pgprw closed map as the image of closed set F= {b, c} in X then f (F)={a, c} in Y which is not pgprw-closed set in Y.

Remark:

The following example show that the β-closed maps and pgprw closed maps are independent and similarly semiclosed maps and pgprw-closed maps.

Example:

Consider X=Y={a,b,c} with topologies τ = {X, ϕ,{a}}and σ ={Y, ϕ,{a}}.Let map f: ( X, τ) (Y, σ) defined by f(a)=b,f(b)=a,f(c)=c.. Then f is pgprw closed map but f is not semi closed map, β-closed map as closed set F= {b, c} in X then f(F)={a,c} in Y which is not semi-closed, β-closed set in Y.

Example:

Consider X= {a, b, c},Y={a, b, c, d} with topologies τ = {X, ϕ,{a}}and σ ={Y, ϕ,{a},{b},{a, b},{a, b, c}}.Let map f: (X, τ) (Y, σ )defined by f(a)=d, f(b)=b, f(c)=c.. Then f is semi-closed map, β-closed map, but f is not pgprw closed map as closed set F= {b, c} in X then f (F)={b, c} in Y which is not pgprw-closed set in Y.

Theorem:

A map f: ( X, τ) (Y, σ) is pgprw-closed map if and only if for each subset S of (Y, σ) and each open set U containing f -1 (S)  U, there is a pgprw-open set of (Y, such that S  V and f -1 (V)  U.

Proof: Suppose f is pgprw-closed map. Let S Y and U be an open set of (X, τ) such that f -1 (S)  U. Now X-U is closed set in (X, τ),. Since f is pgprw-closed map, f (X-U) is pgprw closed set in (Y, σ) . Then V=Y-f (X-U) is a pgprw-open set in(Y, σ). Note that f -1 (S)  U implies S V and f -1 (V) = X-f -1 (f(X-U))  X-(X-U) = U. That is f -1 (V)  U. For the converse, let F be a closed set of ( X, τ) Then f -1 (f(F) c )  F c and F c is an open set in ( X, τ) By hypothesis, there exists a pgprw-open set V in (Y, σ) such that f (F) c V and f -1 (V) F c and so F (f -1 (V)) c . Hence V c  f (F)  f(((f -1 (V)) c ) which implies f(V)  V c . Since V c is pgprwclosed in (Y, σ) and therefore f is pgprw-closed map.

Remark:

The composition of two pgprw-closed maps need not be pgprw-closed map in general and this is shown by the following example. (Y, σ) and g: ( X, τ) (Y, σ) be the identity map. Then f and g are pgprw-closed maps but their composition g of: (X, τ) (Z, ) is not pgprw-closed map because F={c} is closed in ( X, τ) but g of({c}) ={c} which is not pgprw-closed in (Z, ). 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 pgprw-closed map, g (f (F)) is pgprw-closed set in (Z, ). That is g of (F) = g(f(F)) is pgprw-closed and hence g of is pgprw-closed map. (Y, σ) and g:(Y, σ) (Z, )be the identity map. Then f is pgprw closed map and g is a closed map but their composition g of: (X, , τ) (Z, ) is not pgprw-closed map since for the closed set {c}in (X, τ) but g of({c})= {c} which is not pgprw-closed in (Z, ) 3.20 Theorem: Let (X, τ) and (Z, ) be topological spaces and (Y, σ) be topological spac where every pgprw-closed subset is closed. Then the composition g of:(X, τ) (Z, ) of the pgprw-closed maps f: (X, τ) (Y, σ) and g: ( Y, σ) (Z, ) is pgprwclosed map.

3.17

Proof: Let A be a closed set of (X, τ). Since f is pgprw-closed map, f (A) is pgprw-closed map in. Then by hypothesis f (A) is closed. Since g is pgprw-closed map, g (f (A)) is pgprw closed map in (Z, ) and g (f (A)) = g of (A). Therefore g of is pgprwclosed map. Proof: Let A be a closed set of (X, τ) Since f is g-closed map,f (A) is g-closed in (Y, σ),. Since g is pgprw-closed map,g (f (A)) is pgprw-closed in (Z, ) and g (f (A)) = g of (A).Therefore g of is pgprw-closed map.

pgprw-open maps

pgprw-open maps

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