10
Fuzzy Topology

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.

10.1 Metric and Topological Spaces1

On the line images, the distance from images to images is equal to images. More generally, the distance from images to images is denoted by images and it is equal to

equation

Using the notion of distance, we can define continuous functions from images to images as follows:

This distance is called the Euclidean metric.

If we want to be more general, we can talk about any “space” images that has a suitable notion of distance. The new spaces are known as metric spaces.

An ε‐ball about a point images is also called an open ball with center the point images and radius images. The closed ball with center the point images and radius images is the set

equation

A subset images is open if, for each point images, there is an ε‐ball about images completely contained in images. A subset images is closed if its complement (i.e. images) is open. Also, if images and images, then images 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, images, or using a rule, for example images, which means that the imagesth element of the sequence is the number images.

If every Cauchy sequence in images converges to a point in images, then we say that (images is a complete metric space.

The inner product of two elements images and images of some Hilbert space images is written as images.

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.

10.2 Fuzzy Metric Spaces

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 images 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 images of fuzzy sets images that are

  1. normal;
  2. convex;
  3. upper semi‐continuous; and
  4. the closure of images is compact.

Note that, according to the Heine‐Borel theorem, a subspace of images (with the usual topology) is compact if and only if it is closed and bounded.5 These properties imply that for each images, the set images is a nonempty compact convex subset of images, as is the support set images.

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 images‐cuts of the form images, where it is possible to have images and images. Thus, when images, then images is the interval images. The set of all upper semi‐continuous fuzzy numbers will be denoted by images. Also, the set of all nonnegative fuzzy numbers of images is denoted by images.

The arithmetic operations between upper semi‐continuous fuzzy numbers are defined as follows:

equation

for all images. In addition, the neutral elements of addition and multiplication in images are denoted by images and images, respectively, and are defined as follows:

equation

There is a connection between fuzzy normed spaces and fuzzy metric spaces, which is shown by the following result.

The metric images is a fuzzy metric generated by the fuzzy norm images.

10.3 Fuzzy Topological Spaces6

Fuzzy topological spaces have been introduced by C.L. Chang [66].

The pair images 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 images and images are two fuzzy subsets of images. If images, for some images, and if images or images, then images is quasi‐coincident with images. Usually, we write images to denote that images is quasi‐coincident with images.

In order to speak about the neighborhood of a point we need to define fuzzy points.

Given a fuzzy subset images of images, then a fuzzy point images is in images, images, if and only if images. If images, then images is contained in images.

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.

10.4 Fuzzy Product Spaces

Let us first define the product of fuzzy topological spaces.

θ‐topologies are very foundational and important.

One can easily show that

equation

are θ‐topologies.

The proof of the theorem is left as an exercise.

10.5 Fuzzy Separation

Let images be the set of all topologies on images and images the set of all fuzzy topologies on images. On images we consider the topology

equation

Given a topological space images, the induced topology will be denoted by images. We define the mapping images, which maps the family of “functions” images to the initial topology images on images.

Assume that images is a fuzzy topological space and images. Then, the fuzzy set with shape images is called a fundamental fuzzy set for images and is denoted by images.

When images, images, then images is a fuzzy quasi‐images‐space if and only if for every images and images, images.

We state the following result without proof:

It should be clear that a fuzzy images‐space implies a fuzzy images‐space, and a fuzzy images‐space implies a fuzzy images‐space.

A fuzzy set images is a neighborhood of images if and only if images is a neighborhood of every point images, images, and images.

It is easy to show that a fuzzy images‐space is a fuzzy images‐space. Suppose that images is a fuzzy images‐space. Then, any two images are different because of the basic property of fuzzy images‐spaces. Now, we can assume that images, images, and images. Then, there is an open fuzzy set images such that

equation

Assume that images. Then, images is a neighborhood of images and is not quasi‐coincident with images. That is, the fuzzy topological space images is a fuzzy images‐space.

Now we can generalize Urysohn's lemma (see Theorem 10.1.1) on usual topologies to the case of fuzzy topologies.

10.5.1 Separation

10.6 Fuzzy Nets

Assume that images is a set and images is a binary relation on images. Elements images and images are said to be incomparable, written here as images, if neither images nor images is true. Then, the pair images is a semiorder if it has the following properties:

  1. For all images and images, the expressions images and images cannot be simultaneously true.
  2. For all images, images, images, and images, if the expressions images, images, and images are true, then it must also be true that images.
  3. For all images, images, images, and images, if the expressions images, images, and images are true, then it cannot also be true that images and images simultaneously.

10.7 Fuzzy Compactness

As in the case of compact topological spaces, we need to define the notion of a fuzzy cover:

Suppose that images is a fuzzy topological space (à la Chang or à la Lowen).

An alternative definition of à la Chang fuzzy compact spaces follows:

10.8 Fuzzy Connectedness

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.

10.9 Smooth Fuzzy Topological Spaces

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 images, where images and images 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.

10.10 Fuzzy Banach and Fuzzy Hilbert Spaces

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:

equation

10.11 Fuzzy Topological Systems

The Dialectica categories construction (see for example [93]) can be instantiated using any lineale and the basic category images. 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 images, images, and images (images), is a lineale.

Assume that images and images are the following arrows:

equation

Then, images is such that

equation

Tensor products and the internal‐hom in images are given as in the Girard‐variant of the Dialectica construction [89]. Given the objects images and images, the tensor product images is images, where images is the relation that, using the lineale structure of images, takes the minimum of the membership degrees. The linear function‐space or internal‐hom is given by images, where again the relation images is given by the implication in the lineale. With this structure, we obtain:

Products and coproducts are given by images and images, where images is the fuzzy relation that is defined as follows:

equation

Similarly, for the coproduct images.

Steven Vickers [290] introduced topological systems. These structures are triples images, where images is a frame whose elements are called opens and images is a set whose elements are called points. Also, the relation images is a subset of images, and when images, where images and images, we say that images satisfies images. In addition, the following must hold:

  • if images is a finite subset of images, then
    equation
  • if images is any subset of images, then
    equation

Given two topological systems images and images, a map from images to images consists of a function images and a frame homomorphism images, if images. Topological systems and continuous maps between them form a category, which we write as images.

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 images and images, the obvious composition images is also a morphism of fuzzy topological systems. But we know that images is a category, and conditions (i), (ii), and (iii) do not apply to morphisms. Identities are given by images.

The collection of objects of images that are fuzzy topological systems and the arrows between them, form the category images, which is a subcategory of images.

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.

Exercises

  1. 10.1 Explain why the fuzzy metric defined in Example 10.2.1 generalizes the conditions that each crisp metric must satisfy.
  2. 10.2 Show that the triple that is described in Example 10.2.1 is a metric space.
  3. 10.3 Prove that the usual metric space is a special case of the fuzzy metric space defined in Definition 10.2.8.
  4. 10.4 Prove that images in Definition 10.3.15 is a fuzzy topological space.
  5. 10.5 Prove Proposition 10.4.1.
  6. 10.6 Prove Proposition 10.4.2.
  7. 10.7 Prove Proposition 10.4.3.
  8. 10.8 Prove Theorem 10.4.2.
  9. 10.9 Prove Theorem 10.5.5.
  10. 10.10 Prove Theorem 10.5.9.
  11. 10.11 Prove Theorem 10.5.11 for any kind of a fuzzy images‐space, where images.
  12. 10.12 Give a formal definition of smooth topological spaces.
  13. 10.13 Prove Proposition 10.9.2.
  14. 10.14 Show that images from Example 10.10.1 is a fuzzy inner product space.

Notes

  1. 1   The overview of metric spaces and topological spaces that follows is based on [38, 167].
  2. 2   A nonempty class of sets images is called a σ‐ring if and only if
    •  (a) for all images, images and
    •  (b) for all images, images, images.

    A σ‐algebra is a σ‐ring that contains images, where all elements of class images are subsets of images. 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 images is a measurable space (i.e. images is a σ‐algebra). Then, a real‐valued function images on images is measurable if and only if images, for any Borel set images.

  3. 3   The authors defined the Hausdorff distance in [99] using images, while in [100] they defined it using images. Apparently, the definition using images is the correct one.
  4. 4   See Ref. [117].
  5. 5   See Ref. [297].
  6. 6   The text that follows is based on [230] unless explicitly stated otherwise.
  7. 7   Obviously, for a constant fuzzy set images we have that images for all images and images.
  8. 8   A dyadic rational is a fraction whose denominator, when expressed in canonical form (i.e. an expression of the form images, where images and images, that is, they have no common divisor except 1) is an integral power of 2. For example, images or images are dyadic rationals, but images is not.
  9. 9   S&c.breve;ostak did use the term “fuzzy topological spaces,” however, later on the term “smooth fuzzy topological spaces” was put in wide use in the literature and so we have adopted the later term.
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