Some Properties of Soft β-Connected Spaces in Soft Topological Spaces

Soft set theory is a newly emerging tool to deal with uncertain problems and has been studied by researchers in theory and practice. In this paper, we investigated the properties and characterizations of soft -connected spaces in soft topological spaces. We anticipate that the results in this paper can be stimulated to the further study on soft topology to accomplish genenral framework for the practical life applications.


Introduction
The researchers introduced the concept of soft sets to deal with uncertainty and to solve complicated problems in economics, engineering, medicines, sociology and environment because of unsuccessful use of classical methods. The well-known theories that can be considered as mathematical tools for dealing with uncertainties and imperfect knowledge are: theory of fuzzy sets [1], theory of vague sets, theory of interval mathematics [2], theory of intuitionists fuzzy sets [3], theory of rough sets and theory of probability [4,5]. All these tools require the pre specification of some parameter to start with.
In 1999 Molodtsov [6] initiated the theory of soft sets as a new mathematical tool to deal with uncertainties while modeling the problems with incomplete information. In [7], he applied successfully in directions such as, smoothness of functions, game theory, operations research, Riemann-integration, Perron integration, probability and theory of measurement. Maji et al. [8,9] gave first practical application of soft sets in decision making problems.
Shabir and Naz [10] initiated the study of soft topological spaces. They defined basic notions of soft topological spaces such as soft open, soft closed sets, soft subspace, soft closure, soft neighborhood of a point, soft Ti-spaces, for i=1; 2; 3; 4, soft regular spaces, soft normal spaces and established their several properties. In 2011 S. Hussain et al. [11] continued investigating the properties of soft open(closed), soft neighborhood and soft closure. They also defined and discussed the properties of soft interior, soft exterior and soft boundary. Also in the year 2012 B. Ahmad and S. Hussain [12] explored the structures of soft topology using soft points.
A. Kharral et al. [13] defined and discussed the many properties of soft images and soft inverse images of soft sets. They also applied these notions to the problem of medical diagnosis in medical systems. In [14] I. Zorlutana et al, defined and discussed soft pu-continuous mappings. M. Akdag et al. [15] have studied soft b-open sets and soft b-continuous functions, further V. Seenivasan et al. [16] introduced soft gsg-closed sets. I. Arockiarani et al. [17] studied the soft g  -closed sets and soft gs  -closed sets in soft topological spaces. In [18] J. Subhashinin et al. have studied soft P-connectedness in soft topological spaces and Bin Chen [19] continued studying some properties of soft semi-open sets, Recently S.S. Benchalli et al. [20][21][22][23][24] have studied soft  -separation axioms, soft  -compactness, soft  -locally closed spaces, soft regularity, soft normality, weaker forms of soft closed sets and soft  -operations in soft topological spaces. In the present paper new concept in soft topological spaces such as soft  -connected space, soft  -boundary and some of their properties are discussed.
The organization of this paper is as follows: Section 2 briefly reviews some basic concepts about soft sets, soft compactness and related properties in soft topological spaces; Section 3 defines the concepts of soft  -connected spaces and few definitions are given; Section 4 is the main results of the paper, which introduces the soft  -connected and soft  -notconnected spaces and relative properties are studied; Section 5 is conclusion of the paper.

Preliminary
Through-out this paper ) , , ( E X  will be a soft topological space. be two soft sets over a common universe X .
Definition 2.7. [10] Let  be the collection of soft sets over X ; then  is said to be a soft topology on X if it satisfies the following axioms: (1)  , X belong to  .
(2) The union of any number of soft sets in  belongs to  .

Soft  -Connected Spaces in Soft Topological Spaces
In this section, the concepts of soft  -connected space, soft  -boundary and their properties are studied comprehensively.

Results
Theorem 4.1. Let (X, ,E) be a soft topological space. Then the following statements are equivalent: (1) X is soft  -connected.
(2) X and  are the only soft  -clopen sets in (X, ,E).
is non empty and soft  -clopen. So (F, A) and (F, B) are soft  -separated sets. Since Which shows that (5) is false. Therefore and since (F, A) is soft  -connected set then by above theorem 4.2 either ) , . This is contradiction.
Proof: If (F, B) is not soft  -connected, then there exists two soft sets (F, C) and (F, D) such that  .
, it is trivial that . Hence the proof.
. Then by result (1), we have   (2) where (F, B) and (F, C) form a soft  -separation of (F, A) by hypothesis, we may choose a soft point ) , x must belong to either a soft subset of (F, B) or a soft subset of (F, C  is soft  -connected. This completes the proof.

Conclusion
The study of soft sets and soft topology is very important during the study towards applications in non-classical and classical logic. Here, we defined and explored the properties of soft  -connected spaces in soft topological spaces and discussed the behaviour of soft  -connected spaces in soft topological spaces under soft pu-continuous mappings. We have also characterized soft  -connectedness in terms of soft  -boundary and discussed some more properties. We expect that the findings in this paper can be applied to problems of many fields that contains uncertainties and will enhance the further study on soft topology to carry out general framework for the applications in practical life.