Some New Functions in Soft Topological Space

Some New Functions in Soft Topological Space

R. Buvaneswari Department of Mathematics, Alagappa Govt. Arts College

Karaikudi 630 003, Tamil Nadu, India.

A. P. Dhana Balan Department of Mathematics, Alagappa Govt. Arts College

Karaikudi 630 003, Tamil Nadu, India.

Abstract The purpose of this paper is to form some new functions like soft biclop.na-continuous and somewhat soft nearly biclop.na-continuous and also by using these concept, some theorems are analyzed.

Keywords Soft Feebly Open, Soft Feebly Closed, Soft – Open, Soft -Closed.

1. PRELIMINARIES

   Definition 1.1 [4]: Let X be an initial universe set and let E be the set of all possible parameters with respect to X. Let P(X) denote the power set of X. Let A be a nonempty subset of E. A pair (F,A) is called soft set over X, where F is a mapping given by F:AP(X). A soft set (F,A) on the universe X is defined by the set of ordered pairs E,fA(x)P(X)} where fA: EP(X) such that fA(x)= if xA. Here fA is called an approximate function of the soft set (F,A). The collection of soft set (F,A) over a universe X and the parameter set A is a family of soft sets denoted by SS(x)A.

   Definition 1.2[3]: A set set (F,A) over X is said to be null soft set denoted by if for all A,F( e) = . A soft set (F,A) over X is said to be an absolute soft set denoted by A if all A, F( e)=X.

   Definition 1.3[5]: Let Y be a nonempty subset of X, then Y denotes the soft set (Y,E) over X for which Y( e)=Y, for all E. In particular, (X,E) will be denoted by X.

   Definition 1.4 [5]: Let be the collection of soft sets over X, then is said to be a soft topology on X if (i),X (ii)If then (G,E)(iii) If { (Fi,E)}iI then (Fi,E) . The pair (X,,E) is called a soft topological space. Every member of is called a soft open set. A soft set (F,E) is called soft closed in X if .

   Definition 1.5: Let (X,,E) be a soft topological space over X and let (A,E) be a soft set over X

   1. the soft interior [7] of (A,E) is the soft set (A,E)= {(O,E):(O,E) which is soft open

      and(O,E) (A,E)}

   2. the soft closure [5] of (A,E) is the soft set (A,E)

   ={ (F,E) : (F,E) which is soft closed and (A,E) (F,E)}. Clearly (A,E) is the smallest soft closed set over X which contains (A,E) and (A,E) is the largest soft open set over X which is contained in (A,E).

   Definition 1.6 [2]: In a soft topological space (X,,E), a soft set (i) (A,E) is said to be soft feebly-open set if s ( A,E)).

   (ii) (A,E) is said to be soft feebly-closed set if

   s ( (A,E).

   It is said to be soft feebly-clopen if it is both soft feebly-open and soft feebly-closed.

   Definition 1.7 [2]: Let (X,,E) be a soft topological spaces and let (A,E) be a soft set over X.

   1. Soft feebly-closure of a soft set (A,E) in X is denoted by (A,E) = {(F,E): (F,E) which is a soft feebly-closed set and (F,E)}.

   2. Soft feebly-interior of a soft set (A,E) in X is denoted by A,E) = {(O,E) : (O,E) which is a soft feebly-open set and (A,E)}. Clearly (A,E) is the smallest soft feebly-closed set over X which contains (A,E) and A,E) is the largest soft feebly-open set over X which is contained in (A,E).

   Definition 1.8 ([4],[5],[1],[6]) : For a soft (F,E) over the universe U, the relative complement of (F,E) is denoted by (F,E) and is defined by (F,E) = (F,E), where (F,E), where F : EP(U) is a mapping defined by F ( e ) = U F(e) for all E.

2. SOME NEWLY TWO MAPPINGS IN SOFT TOPOLOGICAL

Definition 2.1:Let(A,E) be a subset of soft topological space (X,,E). It is said to be soft -open if for each

x (A,E), there exists an soft open set (G,E) such that x ( (G,E) (A,E). On the other hand

soft -open if for each x A,E), there exists a soft regular open set (U,E) of (X,,E) such that U,E)A,E).

Definition 2.2: Let (A,E) be a subset of soft topological space (X,,E). It is said to be soft -clopen if it is both soft -open and soft -closed.

Remark 2.3: Union of two soft -clopen set is soft -clopen set .

Definition 2.4:A function f : (X,,E) (Y,,E) is said to be soft biclop.na-continuous if the inverse image

of every soft -clopen set (V,E) of Y is soft feeblyclopen in X.

Definition 2.5:Let (A,E) be subset of soft topological space (X,,E). (A,E) is said to be somewhat soft nearly clopen if

( (A,E))) = .

Definition 2.6: A function f : (X,,E) (Y,,E) is somewhat soft nearly biclop.na-continuous if f-1(V,E) is somewhat soft nearly clopen for every soft feebly- clopen set (V,E) in Y such that f-1 (V,E) = .

Theorem 2.7: For a function f : (X,,E) (Y,,E), the following statements are equivalent.

1. f is soft biclop.na-continuous

2. A function f : (X,,E) (Y,,E) is soft biclop.na- continuous if for every soft feebly- clopen set (V,E) of Y containing f(x) there exist soft -clopen set (U,E) containing x such that (V,E).

Proof :(a)(b) : Let X and let (V,E) be a soft feebly- clopen set in Y containing f(x). Then, by (b), f-1(V,E) is soft

-clopen in X containing x. Let (U,E) = f-1(V,E). Then, (V,E).

(b)(a) : Let (V,E) be a soft feebly-clopen set of Y, and let f-1(V,E). Since f(x) (V,E), there exists (U,E) containing x such that f(U,E) V,E). then follows that (U,E) f-1(V,E). Hence f-1(V,E) is soft -clopen.

Theorem 2.8:If f : (X,,E) (Y,,E) and X = X1 X2 where X1 and X2 and soft -copen set and f/X1 and f/X2 are soft biclop.na-continuous, then f is soft biclop.na-continuous.

Proof: Let (A,E) be a soft feebly- clopen subset of Y. Then, since (f/X1) and (f/X2) are both soft biclop.na-continuous, therefore (f/X1)-1(A,E) and (f/X2)-1(A,E) are both soft – clopen set in X1 and X2 respectively. Since X1 and X2 are soft

-clopen subsets of X, therefore (f/X1)-1(A,E) and (f/X2)- 1(A,E) are both soft -clopen subsets of X. Also, f-1(A,E)

= (f/X1)-1(A,E) (f/X2)-1(A,E). Thus f-1(A,E) is the union of two soft -clopen sets and is therefore soft -clopen. Hence f is soft biclop.na-continuous.

Theorem 2.9: If f : (X,,E) (Y,,E)and X = X1 X2 and if (f/X1) and (f/X2) are both soft biclop.na-continuous at a point x belongs to X2, then f is soft biclop.na-continuous at x. Proof: Let (U,E) be any soft feebly- clopen set containing f(x). Since x X1 X2 and (f/X1), (f/X2) are both soft biclop.na-continuous at x, therefore there exist soft -clopen sets (V1,E) and (V2,E) such that x X1 (V1,E) and f(X1 (V1,E)) (U,E), and (X2 (V2,E)) and (X2

(V2,E)) U,E). Now since X = X1 X2, therefore f ((V1,E) (V2,E)) =f(X1 (V1,E) (V2,E)) f(X2 (V1,E)

(V2,E)) (V1,E)) f(X2 (V2,E)) (U,E). Thus,

(V1,E) (V2,E)=(V,E) is a soft -clopen set containing x such that U,E) and hence f is soft biclop.na- continuous at x.

Theorem 2.10: Every restriction of a soft biclop.na- continuous mapping is soft biclop.na-continuous.

Proof:Let f be a soft biclop.na-continuous mapping of (X,,E) into (Y,,E) and t (A,E) be any soft subset of X. For any soft feebly-clopen subset (S,E) of Y, (f/(A,E))-1(V,E) = f-1(V,E). But f is being soft biclop.na-continuous, f- 1(S,E) is soft -clopen and hence f-1(V,E) is a relatively soft -clopen subset of (A,E), that is (f/(A,E))- 1(V,E) is a soft -clopen subset of (A,E). Hence f/(A,E) is soft biclop.na-continuous.

Theorem 2.11: Let f map (X,,E) into (Y,,E) and let x be a point of X. If there exist a soft -clopen set (N,E) of x such that the restriction of f to (N,E) is soft biclop.na-continuous at x, then f is soft biclop.na-continuous at x.

Proof: Let (U,E) be any soft febly-clopen set containing f(x). Since f/(N,E) is soft biclop.na-continuous at x, therefore there is an soft -clopen set (V1,E) such that (N,E) (V1,E) and (V1,E)) (U,E).

Thus (N,E) (V1,E) is soft -clopen set of x.

Theorem 2.12: Let X = (R1,E) (R2,E), where (R1,E), (R2,E)

are soft -clopen sets in X. Let f : (R1,E)(Y,,E) and g : (R2,E)(Y,,E) be soft biclop.na-continuous.

If f(x) = g(x) for each (R1,E) (R2,E). Then

h : (R2,E)(Y,,E) such that h(x) = f(x) for x (R1,E) and h(x) = g(x) for (R2,E) is soft biclop.na-continuous.

Proof:Let (U,E) be a soft feebly- clopen set of Y. Now h-1(U,E) = f-1(U,E) g-1(U,E). Since f and g are soft biclop.na-continuous, f-1(U,E) and g-1(U,E) are soft -clopen set in (R1,E) and (R2,E) respectively. But (R1,E) and (R2,E) are both soft -clopen sets in X. Since union of two soft

-clopen sets is soft -clopen, so h-1(U,E) is a soft -clopen set in X. Hence h is soft biclop.na-continuous.

Theorem 2.13:Let f : (X,,E) (Y,,E) be soft biclop.na- continuous surjection and (A,E) be soft -clopen

subset of X. If f is soft feebly- clopen function, then the function g : (A,E) f(A,E), defined by g(x) = f(x) for each (A,E) is soft biclo.na-continuous.

Proof: Suppose that H = f(A,E). Let (A,E) and (V,E) be any soft feebly- clopen set in (H,E) containing g(x). Since (H,E) is soft feebly clopen set in Y and (V,E) is soft feebly- clopen in (H,E). Since f is soft biclop.na-continuous, hence there exist a soft -clopen set (U,E) in X containing x. Taking (W,E) = (A,E), since (A,E) is soft -open and soft -clopen set in (A,E) containing x. Thus g is soft biclop.na-continuous.

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