Soft pre T 1 Space in the Soft Topological Spaces

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1 International Journal of Fuzzy Mathematics and Systems. ISSN Volume 4, Number 2 (2014), pp Research India Publications Soft pre T 1 Space in the Soft Topological Spaces J. Subhashinin 1 and Dr. C. Sekar 2 1 Department of Mathematics, VV College of Engineering, Tisayanvilai, INDIA. shinijps@gmail. com 2 Department of Mathematics, Aditanar College of Arts and Science, Tirchendur- INDIA. sekar. acas@gmail. com Abstract This paper introduces soft pre T 1 space in the soft topological spaces. The notations of soft pre interior and soft pre closure are generalized using these sets. In a soft topological space, a soft set F A is said to be soft pre-open set (soft P-open) if there exists a soft open set F such that F A F O F. Its complement is known as soft p-closed set of F A denoted by F (F, τ ). A detail study is carried out on properties of soft PT 1 Space. AMS subject classification (2010): 54A10, 54A05 and 06D72. Keywords: Soft pre-open set, Soft pre interior, soft pre closure, soft singleton, soft PT 1 -space. 1. Introduction. Soft set theory was first introduced by Molodtsov [3] in 1999 as a general mathematical tool for dealing uncertain fuzzy, not clearly defined objects. He has shown several applications of this theory in solving many practical problems in economics, engineering, social science, medical science, and so on. Modern topology depends strongly on the ideas of set theory. But in 2010 Muhammad shabir, Munazza Naz [4] used soft sets to define a topology namely Soft topology and defined soft separation axioms. Some of these separation axioms have been found to be useful in computer science and digital topology. In 2013 J. Subhashini and C. Sekar defined soft pre-open sets [7] in a soft topological spaces. In this paper we introduce soft pre separation axioms, soft PT 0 Space and some of its properties.

2 204 J. Subhashinin and Dr. C. Sekar 2. Preliminaries For basic notations and definitions not given here, the reader can refer [1-8] Definition. [4] A soft set F A on the universe U is defined by the set of ordered pairs F = e, f (e) : e E, f (e) P(U), where E is a set of parameters, A E, P (U) is the power set of U, and f : A P(U) such that f (e) = if e A. Here, f is called an approximate function of the soft set F A. The value of f (e) may be arbitrary, some of them may be empty, some may have non-empty intersection. Note that the set of all soft set over U is denoted by S (U) Example. Suppose that there are five cars in the universe. Let U={c 1, c 2, c 3, c 4, c 5 } under consideration, and that E = e, e, e, e, e, e, e, e is a set of decision parameters. Then e (i = 1, 2, 3, 4, 5, 6, 7, 8) stand for the parameters expensive, beautiful, manual gear, cheap, automatic gear, in good repair, in bad repair and "costly respectively. In this case, to define a soft set means to point out expensive cars, beautiful cars and so on. Let A E, the soft set F A that describes the attractiveness of the cars in the opinion of a buyer say Ram, may be defined like A={e, e, e, e, e }, f (e )= {c 2, c 3, c 5 }, f (e )= {c 2, c 4 }, f (e )= {c 1 }, f (e )= {U} and f (e )= {c 3, c 5 }. We can view this soft set F A as consisting of the following collection of approximations: F = e, c, c,, c, (e, {c, c }), (e, {c }), (e, {U}), (e, {c, c }) Definition. [1] Let F be a soft set over U and for the point x U, we say that x F read as x belongs to the soft set F whenever x f (e) for all e A. Not that for any x U, x F if x f (e) for some e A Definition. [1] The soft set F over U such thatf (e) = {x} for all e A is called soft singleton and is denoted by x. It s compliment is denoted by x Definition. [7]Let F S(U). The soft power set of F A is defined by P (F ) = {F : F F, iϵi N} and its cardinality is defined by P (F ) = 2 ( ), where f (e) is the cardinality of f (e) Example. [7] Let U={h 1, h 2 }, E={e 1, e 2, e 3 }, A E, A={e 1, e 2 } and F = { (e, {h, h }), (e, {h, h }) }. Then F = F A, F = F, F ={(e 1, {h 1, h 2 })}, F ={(e 1, {, h 1 })}, F ={(e 1, {h 2 })}, F ={(e 1, {h 1 }), (e 2, {h 1, h 2 })}, F ={(e 1, {h 2 }), (e 2, {h 1, h 2 }}, F ={(e 1, {h 2 }), (e 2, {h 2 })}, F ={(e 1, {h 1 }), (e 2, {h 1 })}, F ={(e 2, {h 1, h 2 })}, F ={(e 2, {h 1 })}, F ={(e 2, {h 2 })}, F ={(e 1, {h 1, h 2 }), (e 2, {h 1 })}, F ={(e 1, {h 1, h 2 }), (e 2, {h 2 })}, F ={(e 1, {h 2 }), (e 2, {h 1 })}, F = {(e 1, {h 2 }), (e 2, {h 2 })} are all soft subset of F A. So P (F ) = 2 4 = Definition. [7] Let F S(U). A soft topology on F denoted by τ is a collection of soft subsets of F having the following properties: (i). F, F τ (ii). F F : i I N τ F ϵ τ (iii). F F : 1 i n, n N τ F ϵ τ.

3 Soft pre T 1 Space in the Soft Topological Spaces 205 The pair (F, τ )or U, τ, E is called a soft topological space Example. Let us consider the soft subsets of F A that are given in Example Then τ = F, F, F, F, τ = F, F, F, τ = {P (F )} are soft topologies on F A Definition [7] Let (F, τ ) be a soft topological space, a soft set F A is said to be soft pre-open set (soft P-open) if there exists a soft open set F such that F A F O F. The set of all soft P-open set of F E is denoted by G (F, τ ) or G (F ). Its complement is known as soft p-closed set of F A denoted by F (F, τ ) Definition [7] Let (F, τ ) be a soft topological space and F F. Then the soft pre closure (soft P-closure) of F A denoted by p(f ) is defined as the soft intersection of all soft P-closed supersets of F A Definition. [8] Let (F, τ ) be a soft topological space over U, F S(U) and Let V be a non null subset of U. Then the soft subset of F over V denoted by vf, is defined as follows:f (e) = V f (e), for all e E. In other words vf = V F Theorem. [7] (i) Arbitrary soft union(intersection) of soft P-open(soft p- closed)sets is a soft P-open(soft p-closed) set. (ii) The soft intersection(union) of any two soft P-open(soft p-closed) set need not be a soft P-open(closed)set. 3. Soft Pre T 1 -space. The soft pre separation axioms are various conditions that are sometimes imposed upon soft topological spaces which can be described in terms of various types of soft separated sets. We have introduced soft pre T 1 (soft PT 1 ) spaces in a soft Topological spaces Definition Let (F, τ ) be a soft topological space over U, and x, y U such that x y. If there exist soft P-open sets F and F such that x F but y F and y F but x F, then (F, τ ) is called a soft pre T 1 space (soft PT 1 -space) Example Consider a soft topological space (F, τ ) where τ = F, F, F, F and consider the soft subset that is given in Example Where U = {h, h }, A = {e, e }, F = {(e, {h, h }), (e, {h, h })} and τ = F, F, F, F. The soft sets are defined as followsf = {(e, {U})}, F = {(e, {U})}. The soft topological space (F, τ ) is also a soft PT 1 - space, Since h, h U and h h such that there exist a soft P-open sets F and F such that h F but h F and h F but h F Example Soft discrete topology is a soft PT 1 -space Example The soft indiscrete topological space is not a soft PT 1 space Example Consider the soft topological space (F, τ ) that is given in Example Where U = {h, h }, E = {e, e, e }, A E, A = {e, e } and F = {(e, {h, h }), (e, {h, h })}. Then τ = F, F, F where F = {(e, {U}), (e, {h })}. Then τ defines a soft topology on U. But this is not a soft PT 1 space because h, h U but there do not exist soft P-open sets F and F such that

4 206 J. Subhashinin and Dr. C. Sekar h F but h F and h F but h F Theorem A soft subspace of a soft PT 1 -space is soft PT 1 -space. Proof: Let (F, τ ) be a soft PT 1 -space over U, and (vf, vτ ) be a soft subspace of (F, τ ) over V. Let x, y V such that x y. Since V U, then x, y U such that x y. Since (F, τ ) is a soft PT 0 space over U. Hence there exist soft P-open sets F and F in U such that x F but y F and y F but x F. Since x V. Then x V. Hence x V F = vf, F is soft P-open set. Consider y F, this implies that y f (e) for some e E. Therefore, y V F = vf. Similarly, if y F but x F, then y vf but x vf, Thus (vf, vτ ) is also a soft PT 1 -space. Therefore the property of being a soft PT 1 -space is a hereditary property Proposition Every soft PT 1 - space is a soft topological space Note Not all soft topological spaces are soft PT 1 -spaces Example Let us consider the soft topological space τ = F, F, F. But by Example 3. 5 it is not a soft PT 1 -space Theorem Let(F, τ ) be a soft PT 1 -space over U iff for eachx U, every soft singleton x of F is a soft P-closed set. Proof: Necessity, Suppose that (F, τ ) is soft PT 1 -space over U and x U. We have to prove that the soft singleton x is soft P-closed or alternatively x is soft P-open set. Let y x, then clearly y x. Now the soft space being soft PT 1 and y x so that there must exist a soft P-open set F such that y F but x F. Thus corresponding to each y x there exist a soft p-open set F such that y F x. Therefore {y: y x} {F : y x} x. or x {F : y x} x. Hence x = {F : y x}. Since F is soft p-open and we know that the arbitrary soft union of soft p-open sets is soft p-open and hence x is soft p-open set. Therefore the soft singleton x is soft p-closed set. Sufficient, Suppose that x U, every soft singleton x of F is a soft P-closed set so thatx is soft P-open set in (F, τ ). We have to prove that (F, τ ) is soft PT 1 -space. Let x, y U and x y such that x and y are soft P-closed and as such x and y are soft pre-open set. Therefore y x but x x and x y buty y. Thus (F, τ ) is soft PT 1 -space over U Proposition A soft topological space (F, τ ) is soft PT 1 -space iff every soft subset of F is soft p-closed set. Proof: Let (F, τ ) be a soft PT 1 -space and x be the arbitrary soft subset of F. Since by Theorem 3. 10, the soft singleton x is a soft p-closed. Now every soft subset of F is the soft union of finite number of soft singletons. Then by Theorem It is soft p-closed. Hence every soft subset of F is soft p-closed set. Conversely, if every soft subset is soft p-closed then as a particular case every soft singleton x is also soft p-

5 Soft pre T 1 Space in the Soft Topological Spaces 207 closd set. Therefore by Theorem the soft space (F, τ ) is a soft PT 1 - space. 4. Conclusion: The initiation of notation of soft topological space was introduced by D. Molodtsov [5] in Many Mathematicians turned their attention to the various concepts of soft topological space. By this way, [6] Muhammad Shabir and MunazzaNaz introduced the concept soft topological spaces. J. Subhashini and C. Sekar [11] introduced the concept of soft pre-open sets. In this paper, we continue this work and introduce soft PT 1 spaces taking help of the soft preopen sets. Also we have studied some related properties. Finally we prove every soft topological space need not be a soft PT 1 Spaces. Reference [1] A. Kandil, O. A. E. Tantawy, S. A. El-Sheikh, A. M. AbD El-LATIF, Soft semi separation axioms and some type of soft functions, Vol. x (x) (2014) [2] H. Maki, J. Umehara and T. Noiri, Every topological space is Pre T 1/2, Mem. Fac. Sci. Kochi Univ. Ser. A, Math., (17) (1996), [3] D. Molodtsov, Soft Set Theory-First Results, Computers and Mathematics with applications, Vol. 37 (1999) [4] Muhammad Shabir and MunazzaNaz, On Soft topological spaces, Computers and Mathematics with Applications, Vol. 63 (2011) [5] A. A. Nasef and A. I. EL-Manghrabi, Further results on pre-t 1/2 Space, Journal of Taibah University for Science Journal Vlo. 2 (2009) Technology, Vol. 55(2) (2013) [6] J. Subhashini, C. Sekar, Soft P-connected via soft P-open sets, International journal of Mathematics trends and Technology, Vol. 6 (2014) [7] J. Subhashini, C. Sekar, Local properties of soft P-open and soft P-closed sets, Proceedings of National Conference on Discrete Mathematic and Optimization Techniques (2014) [8] J. Subhashini, C. Sekar, Soft pre generaliszed-closed sets in a soft topological spaces, International journal of Engineering trends and Technology, Vol. 12 (7) (2014)

6 208 J. Subhashinin and Dr. C. Sekar