Connectedness Degrees in -Fuzzy Topological Spaces
The notion of separatedness degrees of -fuzzy subsets is introduced in -fuzzy topological spaces by means of -fuzzy closure operators. Furthermore, the notion of connectedness degrees of -fuzzy subsets is introduced. Many properties of connectedness in general topology are generalized to -fuzzy topological spaces.
Since Chang  introduced fuzzy theory into topology, many authors have discussed various aspects of fuzzy topology. In a Chang -topology, the open sets are fuzzy, but the topology comprising those open sets is a crisp subset of . However, in a completely different direction, Höhle  presented a notion of fuzzy topology being viewed as an -fuzzy subset of . Then Kubiak  and Šostak  independently extended Höhle's fuzzy topology to -subsets of , which is called -fuzzy topology (see [5, 6]). From a logical point of view, Ying  studied Höhle's topology and called it fuzzifying topology.
Connectivity is one of the most important notions in general topology. It has been generalized to -topology in terms of many forms (see [8–17], etc.). In a fuzzifying topological space, Ying  introduced a definition of connectivity and Fang  proved Fan's theorem. In a -fuzzy topological space , Šostak introduced a notion of connectedness degree by means of the level -topological spaces [20, 21], that is, it can be viewed as connectivity in a -topological space. Although a definition of connectivity was also presented by Yue and Fang  in -fuzzy topological spaces, it was defined for whole -fuzzy topological space not for arbitrary -fuzzy subset.
In this paper, we first introduce the notion of separatedness degrees in -fuzzy topological spaces by means of -fuzzy closure operators. Furthermore, we present the notion of connectedness degrees of -fuzzy subsets, which is a generalization of Yue and Fang's connectedness degree. Many properties of connectedness in general topology can be generalized to -fuzzy topological spaces.
Throughout this paper, denotes a completely distributive DeMorgan algebra. The smallest element and the largest element in are denoted by and , respectively. The set of all nonzero co-prime elements of is denoted by .
We say that is wedge below in , denoted by , if for every subset , implies for some . A complete lattice is completely distributive if and only if for each . For any , define . Some properties of can be found in .
For a nonempty set , the set of all nonzero coprime elements of is denoted by . It is easy to see that is exactly the set of all fuzzy points (). The smallest element and the largest element in are denoted by and , respectively.
For any -fuzzy set and any , we use the following notations:
Definition 2.1 (see [3–5]). An -fuzzy topology on a set is a map such that (LFT1);(LFT2) for all , ;(LFT3) for all , , . can be interpreted as the degree to which is an open set. will be called the degree of closedness of . The pair is called an -fuzzy topological space.
A mapping is said to be continuous with respect to -fuzzy topologies and if holds for all , where is defined by .
Definition 2.2 (see ). An -fuzzy closure operator on is a mapping satisfying the following conditions: (LFC1), for all ;(LFC2) for any ;(LFC3) for any ;(LFC4);(LFC5) for all , . is called the degree to which belongs to the closure of .
Lemma 2.3 (see ). Let be an -fuzzy topological space and let be the -fuzzy closure operator induced by . Then for all , for all ,
3. Separatedness Degrees in -Fuzzy Topological Spaces
In this section, in order to generalize Definition 2.5 to -fuzzy topological spaces, we will introduce the concept of separatedness degrees in -fuzzy topological spaces by means of -fuzzy closure operators.
Definition 3.1. Let be an -fuzzy topological space and . Define Then is said to be the separatedness degree of and .
The following result is obvious.
Proposition 3.2. Let be an -topology on and . Then if and only if and are separated in .
Lemma 3.3. Let be an -fuzzy topological space and . If , then .
Proof. From , we can take such that . Thus we have
Lemma 3.4. Let be an -fuzzy topological space, and . If and , then .
Proof. If and , then and . Hence we have
Lemma 3.5. Let be an -fuzzy topological space, and . Then if and only if there exist such that
Proof. Suppose that . Then for some . This implies
Further more, we have
Hence for any and for any , there are such that , and . Let and . Then, obviously, we have that , , and
Conversely if there exist such that Then by we can obtain that .
4. Connectedness Degrees in -Fuzzy Topological Spaces
Definition 4.1. Let be an -fuzzy topological space and . Define Then is said to be the connectedness degree of .
Proposition 4.2. Let be an -topology on and . Then if and only if is connected in .
Theorem 4.3. Let be an -fuzzy topological space and . Then
Proof. On one hand, we have the following inequality:
on the other hand, in order to prove the inverse, we suppose that (). Then there exist such that and . By Lemma 3.5 we know that there exists such that
Obviously , , , . Hence we have Therefore, The proof is completed.
Example 4.4. Let and . Define by and , and define by and , respectively. Let be defined as follows: Then is an -fuzzy topology on . It is easy to verify that for any and for any .
Corollary 4.5. Let be an -fuzzy topological space. Then
Remark 4.6. Yue and Fang  introduced a definition of connectivity in a -fuzzy topological space. It is easy to see that Yue and Fang's definition is a special case of our definition from Corollary 4.5.
Theorem 4.7. For any , it follows that .
Proof. From Theorem 4.3 we have
Theorem 4.8. For any , one has
Proof. Let and . Now we prove . Suppose that . Then . By Theorem 4.3 we know that there exists such that
By we know that there exists such that . Furthermore by we obtain .
Now we prove . In fact, if , then by we have , hence it follows that contradicting . Analogously, we can prove . Thus by and Theorem 4.3, we know that , contradicting . It is proved that .
Theorem 4.9. For any , one has
Proof. Let and . Now we prove . Suppose that . Then by Theorem 4.3 we know that there exist such that
By we know that one of and must be true.
Suppose that (the case of is analogous). Then we must have otherwise if , then by we know that , contradicting . In this case by we know that . Analogously we can prove . Thus by we can obtain that and . Hence by and Lemma 3.5 we know that , contradicting . This shows that . It is proved that .
By Lemma 3.3 we can obtain the following result.
Corollary 4.10. Let be an -fuzzy topological space and . If , then .
Theorem 4.11. Let be an -fuzzy topological space and . Then
Proof. It is obvious that
Now we prove that
Suppose that (). Take a fixed . Then for any , there exists a such that and . Let . Obviously and . By Corollary 4.10 we easily obtain . This shows
Theorem 4.12. If is continuous, then .
Proof. This can be proved from Theorem 4.3 and the following inequality:
The author would like to thank the referees for their valuable comments and suggestions. The project is supported by the National Natural Science Foundation of China (10971242).
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