 Open Access
 Total Downloads : 145
 Authors : R. Buvaneswari, A. P. Dhana Balan
 Paper ID : IJERTV6IS050248
 Volume & Issue : Volume 06, Issue 05 (May 2017)
 DOI : http://dx.doi.org/10.17577/IJERTV6IS050248
 Published (First Online): 11052017
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
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.nacontinuous and somewhat soft nearly biclop.nacontinuous and also by using these concept, some theorems are analyzed.
Keywords Soft Feebly Open, Soft Feebly Closed, Soft – Open, Soft Closed.

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

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)}

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 feeblyopen set if s ( A,E)).
(ii) (A,E) is said to be soft feeblyclosed set if
s ( (A,E).
It is said to be soft feeblyclopen if it is both soft feeblyopen and soft feeblyclosed.
Definition 1.7 [2]: Let (X,,E) be a soft topological spaces and let (A,E) be a soft set over X.

Soft feeblyclosure of a soft set (A,E) in X is denoted by (A,E) = {(F,E): (F,E) which is a soft feeblyclosed set and (F,E)}.

Soft feeblyinterior of a soft set (A,E) in X is denoted by A,E) = {(O,E) : (O,E) which is a soft feeblyopen set and (A,E)}. Clearly (A,E) is the smallest soft feeblyclosed set over X which contains (A,E) and A,E) is the largest soft feeblyopen 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.


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.nacontinuous 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.nacontinuous if f1(V,E) is somewhat soft nearly clopen for every soft feebly clopen set (V,E) in Y such that f1 (V,E) = .
Theorem 2.7: For a function f : (X,,E) (Y,,E), the following statements are equivalent.

f is soft biclop.nacontinuous

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), f1(V,E) is soft
clopen in X containing x. Let (U,E) = f1(V,E). Then, (V,E).
(b)(a) : Let (V,E) be a soft feeblyclopen set of Y, and let f1(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) f1(V,E). Hence f1(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.nacontinuous, then f is soft biclop.nacontinuous.
Proof: Let (A,E) be a soft feebly clopen subset of Y. Then, since (f/X1) and (f/X2) are both soft biclop.nacontinuous, 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, f1(A,E)
= (f/X1)1(A,E) (f/X2)1(A,E). Thus f1(A,E) is the union of two soft clopen sets and is therefore soft clopen. Hence f is soft biclop.nacontinuous.
Theorem 2.9: If f : (X,,E) (Y,,E)and X = X1 X2 and if (f/X1) and (f/X2) are both soft biclop.nacontinuous at a point x belongs to X2, then f is soft biclop.nacontinuous 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.nacontinuous 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.nacontinuous.
Proof:Let f be a soft biclop.nacontinuous mapping of (X,,E) into (Y,,E) and t (A,E) be any soft subset of X. For any soft feeblyclopen subset (S,E) of Y, (f/(A,E))1(V,E) = f1(V,E). But f is being soft biclop.nacontinuous, f 1(S,E) is soft clopen and hence f1(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.nacontinuous.
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.nacontinuous at x, then f is soft biclop.nacontinuous at x.
Proof: Let (U,E) be any soft feblyclopen set containing f(x). Since f/(N,E) is soft biclop.nacontinuous 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.nacontinuous.
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.nacontinuous.
Proof:Let (U,E) be a soft feebly clopen set of Y. Now h1(U,E) = f1(U,E) g1(U,E). Since f and g are soft biclop.nacontinuous, f1(U,E) and g1(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 h1(U,E) is a soft clopen set in X. Hence h is soft biclop.nacontinuous.
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.nacontinuous.
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.nacontinuous, 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.nacontinuous.
REFERENCES

Bin Chen , Soft semiopen sets and related properties in soft topological spaces, Appl.Math.Inf.Sci.7, No.1: 287294, 2013.

A.P. Dhana Balan and R. Buvaneswari, On Soft Feebly Continuous Functions, Research International Journal of Mathematics and Computer, 2(11): 723728, 2014.

P.K. Maji , R. Biswas and A.R. Roy , Soft set theory,
Comput.Math.Appl.,45:555562, 2003.

D. Molodtsov , Soft set theoryfirst results , Computers and Mathematics with Applications, 37(45):1931, 1999.

M. Shabir and M. Naz, On soft topological spaces,
Comput.Math.Appl., 61:17861799, 2011.

D. Sreeja and C. Janaki, On gbClosed Sets in Topological Space, International journal of Mathematical Archive, Vol 2, 8: 13141320, 2011.

I. Zorlutuna, M. Akdag, W.K. Min and S. Atmaca, Remark on soft topological spaces, Annals of Fuzzy Mathematics and Informatics, 3(2):171185, 2012.