Dynamic Analysis, Learning, and Robust Control of Complex SystemsView this Special Issue
Topological Approaches for Rough Continuous Functions with Applications
In this paper, we purposed further study on rough functions and introduced some concepts based on it. We introduced and investigated the concepts of topological lower and upper approximations of near-open sets and studied their basic properties. We defined and studied new topological neighborhood approach of rough functions. We generalized rough functions to topological rough continuous functions by different topological structures. In addition, topological approximations of a function as a relation were defined and studied. Finally, we applied our approach of rough functions in finding the images of patient classification data using rough continuous functions.
Many studies have appeared recently and dealt with generalizations of topological near-open sets [1, 2] and the possibility of using them in many life applications, including their use in data reduction and reaching some new decisions and conclusions. Rough set theory is a modern approach for reasoning about data [3–7]. This theory depends on a certain topological structure that achieved great success in many areas of real-life applications [8–14]. Now, the general topologists can say, “rough sets theory is a topological bridge from real-life problems to computer science” [15, 16].
Rough set theory was introduced as a novel approach to processing of incomplete data. Among the aims of the rough set theory is a description of imprecise concepts. Suppose we are given a finite nonempty set of elements, called universe. Each element of is characterized by a description, for example, a set of attribute values. In rough sets formulated by Pawlak, an equivalence relation on the universe of elements is determined based on their attribute values. In particular, this equivalence relation is initiated using the equality relation on the attribute values. Many real-world applications have both nominal and continuous attributes [17–19]. It was early recognized that standard rough set model based on the indiscernibility relation is well suited in the case of nominal attributes.
Several procedures were made to overcome limitations of this approach and many authors presented interesting extensions of the initial model (see, for example, [20–24]). It was noted that considering a similarity relation instead of an indiscernibility relation is quite relevant. A binary relation forming classes of objects, which are identical or at least not noticeably different in terms of the available description, can represent the similarities between objects [25–29]. More recent approaches of rough set with its applications can be found in [30–32]. Other rough set theory applications in computer science (field of information retrievals) using topological generalizations can be found in [33–40].
In this paper, we purpose further study on rough functions and introduce new concepts based on rough functions. In Section 2, we give more details regarding the fundamentals of near-open sets. The goal of Section 3 is to introduce the concepts of topological lower and upper approximations of near-open sets and discuss their basic properties. We spotlight on rough numbers in Section 4. We aim in Section 5 to define and study new topological neighborhood approach of rough functions. Section 6 is devoted to generalize the concept of rough function to topological rough function by using different topological structures. Topological approximations of a function as a relation are defined and studied in Section 7. In Section 8, we suggest some applications of rough functions to information systems and give some applications of them in data retrieval. Finally, conclusions of the work are given in Section 9.
2. Basic Concepts of Topological Near-Open Sets
In this part, we recall the definitions of some near-open subsets of a topological space which are useful in the sequel.
A subfamily of the power set of is called a topology if it contains as well as it is closed under arbitrary union and finite intersection. The pair is called a topological space; elements in are called open sets, and their complements are called closed sets.
For a subset of , , and denote respectively the closure, interior, and complement of in , respectively.
A subset of is called,(1)Semi-open (resp., pre-open, open) set if (resp., , ) and its complement is called a semi-closed (resp., pre-closed, closed) set if (resp., , ). A subset which is both semi-open and semi-closed is called semi-regular(2)Semi-pre-open set (or open set) if and it is called a semi-pre-closed set (or closed set) if (3)Regular-open set if and it is called a regular-closed set if (4)-closed set if , where
The α-closure (resp. semi-closure, semi-pre-closure) of a subset of is the intersection of all –closed (resp. semi-closed, semi-pre-closed) sets that contain and is denoted by (resp., , ). The semi-interior of , denoted by , is the union of all semi-open subsets of .
A subset of a topological space is called(1)Generalized closed set if whenever and .(2)Semi-generalized closed (briefly, sg-closed) set if whenever and is semi-open set. Its complement is called a -open set.(3)Generalized semi-closed set if whenever and .(4)-Generalized closed set if whenever and .(5)Generalized α-closed set if whenever and is α- open.(6)-closed set if whenever and is α- open.
3. Topological Near-Open Approach of Rough Approximations
In this section, we introduce and investigate the concepts of topological lower and upper approximations of near-open sets and study their basic properties.
Let be a topological space. If , then(1)Semi-lower approximation of , , where is the family of all semi-open sets in . If we replace the family of all semi-open sets given in (1) above by a family of all pre-open sets (resp., a family of all α- open sets , a family of all -open sets , a family of all regular-open sets , and a family of semi-regular-closed sets ), we obtain pre-lower approximation (resp., α-lower approximation, -lower approximation, regular-lower approximation, and semi-regular-lower approximation).(2)Semi-upper approximation of ,, where is the set of all semi-closed sets in .
If we replace the family of all semi-closed sets given in (2) above by a family of all pre-closed sets (resp., a family of all α-closed sets , a family of all -closed sets , a family of all regular-closed sets , and a family of semi-regular-open sets ), we obtain pre-upper approximation (resp., α-upper approximation, -upper approximation, regular-upper approximation, and semi-regular-upper approximation).
Motivation for topological rough set theory has come from the need to represent subsets of a universe in terms of topological classes of the topological base generated by the general binary relation defined on the universe. That base characterizes a topological space, called topological approximation space, . The topological classes of are also known as the topological granules, topological elementary sets, or topological blocks; we will use to denote the topological class containing . In the topological approximation space, we consider two operators and called the topological lower approximation and topological upper approximation of , respectively. Also, let denote the topological positive region of denotes the topological negative region of , and denotes the topological borderline region of .
The degree of topological completeness characterizes by the topological accuracy measure, in which represents the cardinality of set as follows:
We define here the semi-rough pairs as an example of topological rough sets and we study their properties. You can use any type of the abovementioned near-open sets as another example.
The semi-topological class on a topological approximation space is determined by . A subset is said to be semi-dense (semi-co-dense) if (). By semi-rough pair on , we mean any pair where satisfies the conditions: (Semi-1) is the semi-open set in . (Semi-2) is the semi-closed set in . (Semi-3) . (Semi-4) there is a subset such that(1),(2),(3).
Lemma 1. For any subset in the topological approximation space , the pair is a semi-rough pair on in which every semi-open set in is a semi-closed set.
Proof. Let and . Then, the conditions from (Semi-1) to (Semi-3) are directly satisfied. Now, we need to prove condition (Semi-4). Define , then we have(1)If is a semi-open set, hence that gives which is a contradiction; hence, is not contained in . Then, it must be which gives .(2)Since , , then . Then, we have .(3)Let , , this means that . If , then for every semi-open set and such that implies that and we have , then . If , then there is a semi-open set and . Now, is a semi-open set which contains , and , then there exists a point such that , hence ; therefore, , hence . Then, we have the result .
Lemma 2. For any semi-rough pair in in which every semi-open subset is semi-closed, there are subsets such that and .
Proof. Let be a semi-rough pair and let be any subset, satisfying condition (Semi-4). Define , then , hence . If is a semi-open set, then is another semi-open set contained in . Since , , then , and we have . Therefore, is a semi-open set contained in which means . Since , it follows that and this proves that .
Now, we have . Also, and hence . Then, we have .
Theorem 1. For any topological subspace of the topological approximation space , the function that defined by is bijection.
Proof. First, we will prove that the function is onto as follows: for any semi-rough pair in , then there exists such that . Second, for the proof that is one to one, if , then which implies to and and .
4. Topological Neighborhood Approach of Rough Continuity
Let and be two subsets of a universe , and let and be two approximation spaces, where and are binary relations on and , respectively. We define two subsets and of (also two subsets and of ) which are called right and left neighborhoods of an element . We define now two topologies on and on , respectively, using the intersection of the right and left neighborhoods and as follows:
The rough approximations using these topologies are defined as follows:
The function is called a rough function on if the image of each rough set in is rough in .
Namely, the function f is totally rough iff all subsets , such that , then in .
The function is possibly rough iff some subsets , such that , then in .
Finally, the function is exact iff all subsets , such that , then in .
The function is a topological rough, continuous function on as the following:(1)The function is topological, totally rough, continuous if for all subsets ; if , then in .(2)The function is topological, possibly rough, continuous if for some subsets ; if , then in .(3)Finally, the function is topological exact continuous if for all subsets ; if , then in .
Example 1. Let and be topological spaces, where and and , . Let be a map defined by and , then our results are given in Table 1.
Then, according to Table 1, the function is a topological totally rough continuous function.
Proposition 1. Let and be topological spaces and let be a function. The following are equivalent:(1) is rough continuous.(2)For every , .(3)For every , is a rough continuous at .(4)For every , .
Proof. We will use the sequence (3) implying (1) implying (4) implying (2) implying (3) to prove the equivalence of the proposition. (3) implying (1): suppose a nonempty open set , for a fixed point , we have . But since is rough continuous at , then there exists an open set such that and , then we have and ; this gives that is rough continuous. (1) implying (4): suppose that is rough continuous and let . Let . Let an open set such that . Then, by the definition of rough upper approximation . Let , then . Then, we have . Then, we have . (4) implying (2): fix a closed subset ; let ; we will prove that . But each subset is contained in its upper approximation, . Now, we will prove that . Let , then using (4), we have ; hence or . Then, we have . (2) implying (3): let and be an open set containing . Then, is a closed set and is a closed set in which does not contain the point . But . Then, there exists an open set containing such that , then . Then, is rough continuous at .
Theorem 2. Suppose that be a family of topologies defined on . Let be a rough continuous function for every where is a topological space. Then, is a rough continuous function with respect to the topology .
Proof. Let , then , since is a rough continuous function for every , then . Then, we have in , hence is a rough continuous function with respect to the topology .
Theorem 3. Let be a family of functions. Suppose that is the topology on generated by the class , then(1) is rough continuous for each .(2)If is the intersection of all topologies on such that is rough continuous for each ; then, .(3) is the coarser topology on which gives that is rough continuous for each .(4)The class is a sub-base of .(5)The function is rough continuous if and only if is rough continuous.
Proof. Part (1): for each function if then and . But , then , hence ; then, we have the result. Part (2): we can easily prove that , but the topology is generated by , then . Otherwise, is one of the topologies that make the functions which are rough continuous. Then, we have , hence . Part (3): it is obvious by proof of Part (2). Part (4): since any collection of subsets of is a sub-base of a topology on , then is a sub-base of the topology . Part (5): if the function is rough continuous, then all functions are rough continuous. Otherwise, let be rough continuous and let , then there exists a subset such that . But . Now, we have , then . Then, is rough continuous.
5. Minimal Neighborhood Approach for Rough Continuity
We generalize the concept of rough function to topological rough function by using topological structures. The topological spaces with rough sets are very useful in the field of digital topology which is widely applied in the image processing in computer sciences.
Let be a topological space and . Then, we define
which is called the minimal neighborhood containing the point with respect to the topology on . Let be a topological space, for any element ; we define the subset which is the closure of in .
If is a function between two topological spaces and , we define the functions by for every .
Let be a function, where and are topological spaces. The function is called a topological rough function on if and only if for every . Also, is a topological rough function on if for every point in .
Example 2. Let and be topological spaces, where and and , . Let be a map defined by and , thenThen, we haveAlso,Then, the function is not a topological rough function on and .
A function is said to be topological roughly continuous at the point if and only if , and it is topological roughly continuous on if it is topological roughly continuous at every point .
Example 3. Let be a function defined by and , where and withThen, is a topological rough function on andthen for every , and then is a topological rough continuous function on .
6. Topological Approximations of a Function as a Relation
The function is a relation from to when it satisfies the conditions:(i),(ii)If and , then .
If , we say is a function on . By this way, any function can completely be represented by its graph .
Let be any function, where and are approximation spaces, such that and are any binary relations on and , respectively. We define the relation such that is the blocks of . For the function, , we define the approximations
A function is said to be rough in the approximation space , where and are approximation spaces and , if ; otherwise, is an exact function.
Example 4. Let and be two universes; we define the function , by its graph . Consider the blocks of the binary relations and as follows:Then,Then, we haveTherefore, the function is a rough function such that .
When we have two approximation spaces defined by two equivalence relations, we have the following proposition that governs the product space.
Proposition 2. Let and be two arbitrary approximation spaces. Then, we have .
Proof. Suppose that , and , then we haveSuppose again that .
Then, we haveThen, we have the result as .
Let be any function, where and are arbitrary approximation spaces. We define the relation to be the graph of the function . The rough approximations of are defined as follows:Accordingly, the function is rough if ; otherwise, is an exact function. The pair is called a rough pair of relations.
The following theorems give the conditions on approximation spaces that give exact functions, one-to-one, surjective, and continuous functions.
Theorem 4. The function is an exact function for any selective approximation spaces and .
Proof. The selective approximation space property means that . Then, we have , which yields to that the function is an exact function.
Theorem 5. The function is one-to-one function for any selective approximation spaces and if and only if both and are one-to-one functions.
Proof. The proof is directly using the definition of selective approximation space and using the technology in Theorem 1.
Theorem 6. The function is a surjective function for any selective approximation space and if and only if both and are surjective functions.
Proof. One can prove the theorem using similar technique given in Theorem 1.
Theorem 7. The function is a continuous function for any selective approximation space and if and only if both and are continuous functions.
Proof. As in the technique used in Theorem 5, when we have two topological spaces, generated using two bases , where and are two approximation spaces, then we have the following proposition that governs the product topology.
Proposition 3. Let and be two arbitrary topological spaces. Then, we have .
Proof. Similar to the proof of Proposition 2, the rough pairs of relations satisfied the following two important theorems.
Theorem 8. For the quasidiscrete product topological space , if is a rough pair of relations, and is a subspace of such that is closed in , then is a relative rough pair of relations when .
Proof. The pair is a rough pair of relations in , if the following condition satisfied the following:(1) is an open relation in .(2) is a closed relation in .(3).(4)The relation contains a relation of such that and .Only we need to prove that is a rough pair of relations in ; the proof will end by(1)Since is an open relation in , and is a subspace of , then is an open relation in .(2)Since is a closed relation in , then there is an open relation , such that , then , but is an open relationship with , then is a closed relation in the subspace .(3)Since , then .(4)By selecting , then the relation contains the relation , and we need to prove the two subconditions:(a),(b).For the proof of Part (a) , suppose that , then there is an -open relation such that but , i.e., , but such that is an open relation in , then ; hence, , but , which gives contradiction; then, it must be .
For the proof of Part (b), .
Since is a rough pair in , then there is a relation , such that and ; since , we have .
Now, let , then and .
Now, if , then and .
Finally, if and and , hence and