Geometry and topology are two different branches of mathematics that deal with the same objects. However, as K.D. Joshi [167, p. 67] notes “…if two objects are equivalent for a geometer, they are certainly so for a topologist” and “…two objects which look distinct to a geometer may look the same to a topologist.” In this chapter, we discuss fuzzy topology, and in the next chapter, we explore fuzzy geometry. Since metrics induce topologies, we will start with fuzzy metric spaces.
On the line , the distance from to is equal to . More generally, the distance from to is denoted by and it is equal to
Using the notion of distance, we can define continuous functions from to as follows:
This distance is called the Euclidean metric.
If we want to be more general, we can talk about any “space” that has a suitable notion of distance. The new spaces are known as metric spaces.
An ε‐ball about a point is also called an open ball with center the point and radius . The closed ball with center the point and radius is the set
A subset is open if, for each point , there is an ε‐ball about completely contained in . A subset is closed if its complement (i.e. ) is open. Also, if and , then by the third property of the previous definition. This shows that all ε‐balls are open sets. The following proposition shows that for continuity we need only open sets:
A sequence is a list of objects (usually numbers) that are in some order. Typically, we can write its elements like a set, that is, , or using a rule, for example , which means that the th element of the sequence is the number .
If every Cauchy sequence in converges to a point in , then we say that ( is a complete metric space.
The inner product of two elements and of some Hilbert space is written as .
Since continuity can be expressed in terms of open sets alone, and since some interesting constructions of spaces do not rely on the metric, it is very useful to forget about metrics and to focus on the basic properties of open sets that are required in order to talk about continuity. Quite naturally, this leads to the notion of a general topological space.
The following can be found in [219]:
Continuity is a basic notion in topology and neighborhoods can also be used to define continuity:
The following definition is borrowed from [38, p. 7]:
The so‐called separation axioms are used to describe topological spaces that have further restrictions in addition to the “standard” ones.
A normal space has a number of important properties. One of them is that normal spaces admit a lot of continuous functions:
In a connected space, it is possible to go from one point to any other point without jumps. Intuitively, one could say that if Greece and Switzerland were topological spaces, then Greece is not connected while Switzerland is connected.
Fuzzy metric spaces were introduced by Ivan Kramosil and Jiří Michálek [181]. In their approach, the distance between two points is not just a real number but something vague. For example, a real number with some plausibility degree. In what follows, the set contains such degrees.
This lemma leads to the definition that follows.
A simplified formulation of this definition has been provided in [134]:
Before presenting another approach to the definition of fuzzy metric spaces, which was formulated by Phil Diamond and Peter Kloeden [99, 100], we need to give some definitions.
The definition that follows is borrowed from [270, p. 120]
Alternatively, one can use the definition that follows.4
Diamond and Kloeden examined the class of fuzzy sets that are
Note that, according to the Heine‐Borel theorem, a subspace of (with the usual topology) is compact if and only if it is closed and bounded.5 These properties imply that for each , the set is a nonempty compact convex subset of , as is the support set .
We state the following result without proof:
Osmo Kaleva and Seppo Seikkala [171] introduced their own version of fuzzy metric space. The distance between two points in this alternative fuzzy metric space is a nonnegative, upper semi‐continuous fuzzy number. An upper semi‐continuous fuzzy number has ‐cuts of the form , where it is possible to have and . Thus, when , then is the interval . The set of all upper semi‐continuous fuzzy numbers will be denoted by . Also, the set of all nonnegative fuzzy numbers of is denoted by .
The arithmetic operations between upper semi‐continuous fuzzy numbers are defined as follows:
for all . In addition, the neutral elements of addition and multiplication in are denoted by and , respectively, and are defined as follows:
There is a connection between fuzzy normed spaces and fuzzy metric spaces, which is shown by the following result.
The metric is a fuzzy metric generated by the fuzzy norm .
Fuzzy topological spaces have been introduced by C.L. Chang [66].
The pair is called a fuzzy topological space.
Robert Lowen [197] has given an alternative definition of fuzzy topology that is based on his work on fuzzy‐compactness:
In what follows, we will use Chang's definition since it is the one that is used the most in the literature.
Bases and sub‐bases are defined as follows:
In Lowen's [197] version, bases and sub‐bases are defined as follows:
Chang defined the neighborhood of a fuzzy set as follows:
Note that here we are not talking about the neighborhood of a point. Assume that and are two fuzzy subsets of . If , for some , and if or , then is quasi‐coincident with . Usually, we write to denote that is quasi‐coincident with .
In order to speak about the neighborhood of a point we need to define fuzzy points.
Given a fuzzy subset of , then a fuzzy point is in , , if and only if . If , then is contained in .
The definition that follows is from [300].
The interior and the closure of a fuzzy set in Lowen's [197] formulation are defined as follows:
Essentially, the two definitions are similar.
A complete theory of fuzzy topological spaces should include a definition of continuity.
We conclude this section with a definition of subspaces.
Let us first define the product of fuzzy topological spaces.
θ‐topologies are very foundational and important.
One can easily show that
are θ‐topologies.
The proof of the theorem is left as an exercise.
Let be the set of all topologies on and the set of all fuzzy topologies on . On we consider the topology
Given a topological space , the induced topology will be denoted by . We define the mapping , which maps the family of “functions” to the initial topology on .
Assume that is a fuzzy topological space and . Then, the fuzzy set with shape is called a fundamental fuzzy set for and is denoted by .
When , , then is a fuzzy quasi‐‐space if and only if for every and , .
We state the following result without proof:
It should be clear that a fuzzy ‐space implies a fuzzy ‐space, and a fuzzy ‐space implies a fuzzy ‐space.
A fuzzy set is a neighborhood of if and only if is a neighborhood of every point , , and .
It is easy to show that a fuzzy ‐space is a fuzzy ‐space. Suppose that is a fuzzy ‐space. Then, any two are different because of the basic property of fuzzy ‐spaces. Now, we can assume that , , and . Then, there is an open fuzzy set such that
Assume that . Then, is a neighborhood of and is not quasi‐coincident with . That is, the fuzzy topological space is a fuzzy ‐space.
Now we can generalize Urysohn's lemma (see Theorem 10.1.1) on usual topologies to the case of fuzzy topologies.
Assume that is a set and is a binary relation on . Elements and are said to be incomparable, written here as , if neither nor is true. Then, the pair is a semiorder if it has the following properties:
As in the case of compact topological spaces, we need to define the notion of a fuzzy cover:
Suppose that is a fuzzy topological space (à la Chang or à la Lowen).
An alternative definition of à la Chang fuzzy compact spaces follows:
In what follows, we will briefly describe the notion of fuzzy connectedness.
The proof of the following result is omitted since it is based on a number of results that we have intentionally omitted – obviously, we did not plan to cover the whole field of fuzzy topology in a chapter.
In Section 10.3, we presented two different formulations of fuzzy topological spaces. Both of these formulations are crisp, mainly because fuzzy sets either belong or do not belong to the collection of fuzzy open sets. However, this seems a bit unnatural and so S&c.breve;ostak [264] introduced his own version of fuzzy topological spaces where an open set is open to some degree.
If we assume that , where and are lattices, then we get more general structures that are called smooth topological spaces. These structures have been introduced by Ahmed A. Ramadan [246]. However, there are many generalizations of fuzzy topological spaces and we had no intention to cover all of them. After all, it is quite possible to define any kind of extension.
The following result can be easily proved using the definitions above.
In simple words, a mapping is a fuzzy continuous mapping if it does not decrease the degree of openness of fuzzy subsets in the opposite direction (i.e. the direction of the preimage operator).
The proof is obvious and so it is omitted.
In this section, we introduce some topological notions of fuzzy normed spaces (see Section 9.6).
Fuzzy Hilbert spaces have been introduced by Manizheh Goudarzi and Vaezpour [147]. In what follows, we will make use of the following real step function:
The Dialectica categories construction (see for example [93]) can be instantiated using any lineale and the basic category . A lineale is a structure defined as follows:
As was discussed in [272], the unit interval, since it is a Heyting algebra, has all the properties of a lineale structure. In particular, one can prove that the quintuple , , and (), is a lineale.
Assume that and are the following arrows:
Then, is such that
Tensor products and the internal‐hom in are given as in the Girard‐variant of the Dialectica construction [89]. Given the objects and , the tensor product is , where is the relation that, using the lineale structure of , takes the minimum of the membership degrees. The linear function‐space or internal‐hom is given by , where again the relation is given by the implication in the lineale. With this structure, we obtain:
Products and coproducts are given by and , where is the fuzzy relation that is defined as follows:
Similarly, for the coproduct .
Steven Vickers [290] introduced topological systems. These structures are triples , where is a frame whose elements are called opens and is a set whose elements are called points. Also, the relation is a subset of , and when , where and , we say that satisfies . In addition, the following must hold:
Given two topological systems and , a map from to consists of a function and a frame homomorphism , if . Topological systems and continuous maps between them form a category, which we write as .
Fuzzy topological systems have been introduced by Syropoulos and de Paiva [279].
To see that fuzzy topological systems also form a category, we need to show that, given morphisms and , the obvious composition is also a morphism of fuzzy topological systems. But we know that is a category, and conditions (i), (ii), and (iii) do not apply to morphisms. Identities are given by .
The collection of objects of that are fuzzy topological systems and the arrows between them, form the category , which is a subcategory of .
The following result is based on the previous one:
Obviously, it is not enough to provide a generalization of structures – one needs to demonstrate that these new structures have some usefulness. The following example provides an interpretation of these structures in a “real‐life” situation.
A σ‐algebra is a σ‐ring that contains , where all elements of class are subsets of . Consider the class of all bounded, left closed, and right open intervals of the real line. These sets have certain properties and are known as Borel sets. Assume that is a measurable space (i.e. is a σ‐algebra). Then, a real‐valued function on is measurable if and only if , for any Borel set .
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