Fuzzy Soft Connected Sets in Fuzzy Soft Topological Spaces I

In this paper we introduce some types of fuzzy soft separated sets and study some of thier preperties. Next, the notion of connectedness in fuzzy topological spaces due to Ming and Ming, Zheng etc., extended to fuzzy soft topological spaces. The relationship between these types of connectedness in fuzzy soft topological spaces is investigated with the help of number of counter examples.


INTRODUCTION
After Zadeh [26] introduced the notion of a fuzzy set in 1965, Chang [4] used that concept to define fuzzy topology. In 1999, Molodtsov [15] introduced the concept of soft set theory which is a completely new approach for modeling uncertainty. In [15], Molodtsov established the fundamental results of this new theory and successfully applied the soft set theory into several directions, such as smoothness of functions, operations research, Riemann integration, game theory, theory of probability and so on. Maji et al. [13] defined and studied several basic notions of soft set theory in 2003. Shabir and Naz [21] introduced the concept of soft topological space.
Maji et al. [12] initiated the study involving both fuzzy sets and soft sets. In this paper, the notion of fuzzy soft sets was introduced as a fuzzy generalizations of soft sets and some basic properties of fuzzy soft sets are discussed in detail. Maji et al. combined fuzzy sets and soft sets and introduced the concept of fuzzy soft sets. To continue the investigation on fuzzy soft sets, Ahmad and Kharal [1] presented some more properties of fuzzy soft sets and introduced the notion of a mapping on fuzzy soft sets. In 2011, Tanay et al. [22] gave the topological structure of fuzzy soft sets.
In this paper, we extend the notion of connectedness of fuzzy topological space to fuzzy soft topological space. In Section 3, we introduce the different notions of fuzzy soft separated sets and study the relationship between them. Section 4 is devoted to introduce the different notions of connectedness in fuzzy soft topological space and study the implications that exist between them. Also, we study the characterization of connectedness in fuzzy soft setting.

PRELIMINARIES
Throughout this paper denotes initial universe, denotes the set of all possible parameters which are attributes,characteristic or properties of the objects in , and the set of all subsets of will be denoted by . In this section, we present the basic definitions and results of soft set theory which will be needed in the sequel. Aktaş and Ça man [3] showed that every fuzzy set may be considered as a soft set. That is, fuzzy sets are a special class of soft sets. (2) is called fuzzy soft discrete topology on .
Note that, the intersection of any family of fuzzy soft topologies on is also a fuzzy soft topology on .   Definition 2.23. [9] Let Otherwise, is called -disconnected set for .
In the above definition, if we take instead of , then the fuzzy soft topological space is called connected space .

Remark 2.2. [9]
The relationship between -connectedness can be described by the following diagram:

Remark 2.3. [9]
The reverse implications is not true in general (see Examples 3.14, 3.15, 3.16, 3.17 in [9]). But example 3.17 in [9] is incorrect, we must take if Theorem 2.5. [9] Let be a fuzzy soft surjective continuous mapping and . If is a -connected set in , then is a -connected set in for

FUZZY SOFT SEPARATED SETS IN FUZZY SOFT TOPOLOGICAL SPACES
In this section, we will introduce different notions of fuzzy soft separated sets and study the relation between these

Example3.2. Let
, and . 0 and are fuzzy soft separated sets. But and are not fuzzy soft -separated sets since,       Proof. As a sample, we will prove the case fuzzy soft -sepaeated. Let and be fuzzy soft -separated sets in . Then, Since and , then

FUZZY SOFT CONNECTED SETS IN FUZZY SOFT TOPOLOGICAL SPACES
In this section, we introduce different notions of connectedness of fuzzy soft sets and study the relation between these notions. Also, we will investegate the characterizations of the fuzzy soft connected sets.. . Note that, this kind of fuzzy soft connectedness was studied by Mahanta [11].
In the above definitions, if we take instead of , then the fuzzy soft topological space is called - is not a -connected. If we take and then and .
Therefore, and are not fuzzy soft weakly separated sets. Hence, is a -connected.
, then which whould imply (since ). This is a contradiction. Similarly, Therefore, is a -disconnected. So, is a -connected.
is not a -connected which is a contradiction.
Conversely, let be a -connected. Suppose is not a -connected. There exist two non-null fuzzy soft Again, for every . Therefore, is not a -connected. This is a contradiction. So, is a -connected.

Remark 4.5. If is a
-connected, then it may not be a -connected as shown by the following example.       Proof. As a sample, we will prove the case . Let be a -connected set in . Suppose, is not a -connected set in . Then, has a non-null proper clopen fuzzy soft subset .
So, there exist and such that Also, since and and is a fuzzy soft continuos function, then and .
Hence, is a non-null proper clopen fuzzy soft subset of which is a contradiction. Therefore, is a -connected set in . Hence, and so This complete the proof. Therefore, Now, consider the following cases: We claim that,  . For if not, then Now by (3) and (4)