On Contra –rgα-Continuous Functions Types And Almost Contra –rgα-Continuous Function

: Abstract The main aim of this paper is to study some new classes of contra – continuous funtions which are (contra –rgα-continuous function ,contra –rgα*- continuous function and contra–rgα**-continuous function ) in topological space and introduce some of their properties and relation among them .Also we define and study other type of contra-rgα-continuous function called almost contra –rgα-continuous function .


1α-open set if A int(cL(int(A)) and α-closed set if cL(int(cL(A))  A 1regular open set if A=int(cL(A)) and regular closed set if A=cL(int(A))
The intersection of all α-closed subset of (X ,τ) containing A is called the αclosure of A and is denoted by αcL(A) .

Definition(2-8): [6]
A subset A of a topological space(X, τ) is said to be Regular generalized closed (briefly, rg-closed)if cL(A)  U whenever A U and U is regular open set in X.

Definition(2-7): [6] :
A space (X,τ) is said to be a T * ½-space if every rg-closed set in X is closed.
Example(2-8): Consider X={a,b,c } with topology τ ={X,ø ,{a} ,{a,c }}.let the function f: (X, τ) →(X, τ) be defined by f(a)=b, f (b)=c and f(c)=a.It is clear that f is contra-α -continuous but is not contracontinuous function .since the set {a} is an open set in X.But f -1 (a)=c is αclosed but is not closed set in (X, τ) .

8-On Contra-rgα-Continuous Functions Types:
In this section we introduce and study new class of contra-continuous function namely (contra-rgα-continuous , contra-rgα * -continuous and contra-rgα ** - continuous)function and discussion the relations between them.Also, we give some proposition and results about the composition of these functions.

Proof:
It follows from the definition (1-1) and the fact that every closed (resp.αclosed) set is rgα-closed set.

Here, we define and introduce some new notions and results, that shall needed in this work. Definition(8-8):
A space (X,τ)is said to be rgα-locall indiscrete if every rgα-open set in X is closed.

Proof:
Let A be an rgα-open set in (X,τ).Then A c is a rgα-closed set in (X,τ), Since X is a T * ½-space.Then A c is a closed set in X .Hence A is an open set in X.

Proof:
Let A is an open set in Y. Thus f -1 (A) is a rgα-open set in X .Since X is rgαlocally indiscrete and by using definition (3-4) we get f -1 (A) is a closed set in X and by Remark (1-4) we have f -1 (A) is a rgα-closed set in X. Hence ,f is contrargα-continuous.

Now, the following proposition and corollary given the condition to make the converse of a proposition (8-8) true. Proposition(8-11):
If a function f: (X, τ) →(Y, σ) is contra-rgα-continuous and X is T * ½ space ,then f is contra-continuous.

Proof:
Let A be an open set in Y. Thus, f -1 (A) is a rgα-closed set in X .sinceX is a T * ½ space and by using proposition (3-5) we get f -1 (A) is a closed set in X .Hence f is contra -continuous function.

Proof:
Assume that f is contra-rgα * -continuous.Let A be any rgα-open set in (Y,σ).
is an open set in (X, τ) .Therefore f is contra-rgα * -continuous function .
Proposition(8-15): Every contra-rgα * -continuous function is contra-continuous.Proof: Let f: (X τ) →(Y ,σ) be a contra-rgα * -continuous function and let A be an open set in Y .By Remark(1-4) we get A is rgα-open set in Y. Thus f -1 (A) is A closed set in X.Hence a function f is a contra -continuous.

Here, in the following proposition and result given the condition to make the converse of a proposition(8-15) and corollary(8-16) true. Proposition(8-17): If f: (X ,τ)→(Y,σ) is contra -continuous function and let Y is T * ½ space
Then f is contra-rgα * -continuous .

Proof:
Let A be an rgα-open set in Y. since Y is T * ½space and by using corollary(3-1) we get A is an open set in Y.Since f is a contra-continuous function.Thus, f - 1 (A c ) is a closed set in (X,τ).Hence, a function f is a contra-rgα * -continuous.

Proposition(8-19):
If f: (X, τ) →(Y ,σ) is a strongly -rgα-continuous function and let X is a locally indiscrete.Then f is contra-rgα * -continuous.Proof: But X is a rgα-locally indiscrete.Then f -1 (A) is a closed set in X.Hence, f is contrargα * -continuous function.Suppose that f is contra-rgα ** -continuous function and let A be any rgα-open set in (Y,σ).Then A c is a rgα-closed set in (Y,σ).Thus,

Proof:
Let f: (X,τ) →(Y,σ) be a contra-rgα ** -continuous function and let A is a closed set in Y and by using Remark(1-4) we get A is a rgα-closed set inY.Since f is a contra-rgα ** -continuous.Then f -1 (A) is a rgα-open set in X. Hence ,a function f is a contra-rgα-continuous.But , the converse of above proposition need not be true as seen from the following example.Example(8-8): Let X= {a ,b, c,d } with topology τ={X ,ø ,{a}, {b },{a,b},{b,c},{a,b,c},{a,b,d}}.Let the function f: (X,τ) →(X,τ) be defined by f(a)= f(c)=a , f(b)=d , and f(d)=b .Then this function is contrargα-continuous But is not contra-rgα ** -continuous function, since the set {a,b }is rgα-closed set in X but f -1 ({a,b})={a,c,d } is not rgα-open set in X.

‫مجلة‬
Since the {a,c }is an rgα-open set in X but f 1 ({a,c}) ={a,c } is rgα-closed set in X but is not closed set.

Proof:
Let A be a rgα-closed set in Y. Since Y is T * ½ space and by proposition(3-5) we get A is a closed set in Y and also since f is a contra-rgα-continuous.Thus, f -1 (A) is a rgα-open set in X.Hence, f is contrargα ** -continuous.Similarly ,we prove the following corollary.Proposition(8-27): If f: (X,τ) →(Y,σ) is a contra-rgα ** -continuous function and let X be a T * ½ space.Then f is contrargα * -continuous.

Remark(8-23):
The following example shows that contra-continuous function and contra-αcontinuous function are independent of a contra-rgα ** -continuous function.

1 -
briefly rα-open) if there is a regular open set U such that U A αcL(U).The family of all regular α-open sets of X is denoted by Rα0(X).regular generalized α-closed set (briefly, rgα-closed) If αcL(A)  U whenever A U and U is regular αopen in X.The set of all rgα-closed set in X denoted by RG αC(X) .3-regular generalized α-open (briefly, rgα-open) in X If A c is rgα-closed set in X .The family of all rgα-open sets in X denoted by RG αO(X).

1 -
Every open (resp.closed ) set in α-open (resp.α-closed) set in X. 1-Every α-open (resp.α-closed) set is rgα-open (resp.rgα-closed)set in X.3-The union of two rgα-closed subset of X is also rgα-closed subset of X.But the intersection of two rgα-closed set in X is generally not rgα-closed set in X .But, the converse of Remark (2-5) need not be true, as seen from the following example Let X={a,b,c } with topology τ ={X,ø ,{a} ,{a,c }}.Then the set A={a,b} is α-open set but is not open set in X and A c ={c} is a αclosed set in X but is not closed set in X.Also ,the set B={b,c }is a rgα-open set but is not α-open set in X and B c ={a} is a rgα-closed set in X but is not αclosed .Definition(2-6): [5] ,[3], [11]A function f: (X ,τ) →(Y, σ) is said to be 1-continuous if the inverse image of every open (closed) set in Y is an open (closed )set in X. 1-α-continuous if the inverse image of every open (closed) set in Y is an αopen (α-closed) set in X. 3rgα-continuous if the inverse image of every open (closed)set in Y is an rgαopen (rgα-closed )set in X. 4rgα-irresolute if the inverse image of every rgα-open (rgα-closed) set in Y is an rgα-open(rgα-closed) set in X. 5-Strongly rgα-continuous if the inverse image of every rgα-open(rgα-closed) set in Y is an open (closed)set in X. 1-almost continuous if the inverse image of every regular open set in Yis an open set in X.

converse of the above Remark need not be true ,as seen from the following example. Example(2-1): Let
1-Every open (resp.closed ) set in X is rgα-open (resp.rgα-closed)set.1-Every rgα-closed (resp.rgα-open)set in X is rg-closed (resp.rg-open)set.The X={a,b,c,d,e} with topology τ ={X,ø ,{a} , {d} , {e} ,{a,d},{a,e} ,{d,e} ,{a,d,e}} .Then the set A={b} is a rgα-closed set but is not closed set in X ,and the set B={a,b} is rg-closed set but not rgα-closed set in X.alsoA c =,{a,c,d} is a rgα-open set but is not open in X,and B c ={c,d} is a rg -open set but is not rgα-open.