`Abstract and Applied AnalysisVolume 2011, Article ID 762361, 41 pageshttp://dx.doi.org/10.1155/2011/762361`
Review Article

## On Random Topological Structures

1Department of Mathematics, Florida Institute of Technology, FL 32901, USA
2Department of Mathematics Education and the RINS, Gyeongsang National University, Chinju 660-701, Republic of Korea
3Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran

Received 27 January 2011; Accepted 6 April 2011

Copyright © 2011 Ravi P. Agarwal et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We present some topics about random spaces. The main purpose of this paper is to study topological structure of random normed spaces and random functional analysis. These subjects are important to the study of nonlinear analysis in random normed spaces.

#### 1. Introduction

The purpose of this paper is to give a comprehensive text to the study of nonlinear random analysis such as the study of fixed point theory, approximation theory and stability of functional equations in random normed spaces. The notion of random normed space goes back to Šherstnev [1] and Hadžić [24] who were dulled from Menger [5], and Schweizer and Sklar [4] works. Some authors [611] considered some properties of probabilistic normed and metric spaces also fuzzy metric and normed spaces [1221]. Fixed point theory [3, 2226], approximation theory [2731], and stability of functional equations [3238] are studied at random normed space and its depended space that is, fuzzy normed space.

This paper is introduced as a survey of the latest and new results on the following topics.(i)Basic theory of triangular norms.(ii)Topological structure of random normed spaces.(iii)Random functional analysis.(iv)Relationship between random normed spaces and fuzzy normed spaces.

#### 2. Triangular Norms

Triangular norms first appeared in the framework of probabilistic metric spaces in the work Menger [5]. It also turns out that this is a crucial operation in several fields. Triangular norms are an indispensable tool for the interpretation of the conjunction in fuzzy logics [39] and, subsequently, for the intersection of fuzzy sets [40]. They are, however, interesting mathematical objects themselves. We refer to some papers and books for further details (see [2, 4, 4144]).

Definition 2.1. A triangular norm (shorter -norm) is a binary operation on the unit interval , that is, a function such that for all the following four axioms are satisfied:
(T1) (commutativity);
(T2) (associativity);
(T3) (boundary condition);
(T4) whenever (monotonicity).

The commutativity of (T1), boundary condition (T3), and the the monotonicity (T4) imply that for each -norm and for each the following boundary conditions are also satisfied: and therefore all -norms coincide on the boundary of the unit square .

The monotonicity of a -norm in its second component (T3) is, together with the commutativity (T1), equivalent to the (joint) monotonicity in both components, that is, to Basic examples are the Łukasiewicz -norm : and the -norms ,,, where

If, for two -norms and , the inequality holds for all , then we say that is weaker than or, equivalently, that is stronger than .

As a result of (2.2) we obtain, for each , Since, for all , trivially , we get for an arbitrary -norm , That is, is weaker and is stronger than any other -norms. Also since , we obtain the following ordering for four basic -norms

Proposition 2.2 (see [2]). (i) The minimum is the only -norm satisfying for all .
(ii) The weakest -norm is the only -norm satisfying for all .

Proposition 2.3 (see [2]). A -norm is continuous if and only if it is continuous in its first component, that is, if for each the one-place function is continuous.

For example, the minimum and Łukasiewicz -norm are continuous but the t-norm defined by for , is not continuous.

Definition 2.4. (i) A -norm is said to be strictly monotone if (ii) A -norm is said to be strict if it is continuous and strictly monotone.

For example, the -norm is strictly monotone but the minimum and Łukasiewicz -norm are not.

Proposition 2.5 (see [2]). A -norm is strictly monotone if and only if

If is a t-norm, then is defined for every and by 1, if and , if .

Definition 2.6. A -norm is said to be Archimedean if for all there is an integer such that

Proposition 2.7 (see [2]). A -norm is Archimedean if and only if for each one has

Proposition 2.8 (see [2]). If -norm is Archimedean, then for each one has

For example, the product , Łukasiewicz -norm , and weakest -norm are all Archimedean but the minimum is not an Archimedean -norm.

A t-norm is said to be of Hadžić-type (we denote by ) if the family is equicontinuous at , that is, is a trivial example of a t-norm of Hadžić type, but is not of Hadžić type.

Proposition 2.9 (see [2]). If a continuous -norm is Archimedean, then it cannot be a -norm of Hadžić type.

Other important triangular norms are [see [45]]:(i)the Sugeno-Weber family is defined by and if ;(ii)the Domby family , defined by , if , if and if ;(iii)the Aczel-Alsina family , defined by , if , if and

if .

A -norm can be extended (by associativity) in a unique way to an -array operation taking for any the value defined by

can also be extended to a countable operation taking for any sequence in the value The limit on the right side of (2.20) exists since the sequence is nonincreasing and bounded from below.

Proposition 2.10 (see [45]). (i) For , the following implication holds:
(ii) If is of Hadžić-type, then for every sequence in such that .
(iii) If , then
(iv) If , then

Definition 2.11. Let and be two continuous -norms. Then dominates , denoted by , if for all ,

Now, we extended definitions and results on the triangular norm on lattices. Let be a complete lattice, that is, a partially ordered set in which every nonempty subset admits supremum and infimum, and , .

Definition 2.12 (see [46]). A -norm on is a mapping satisfying the following conditions:
(a) (boundary condition);
(b) (commutativity);
(c) (associativity);
(d) and (monotonicity).

Let be a sequence in converges to (equipped order topology). The -norm is said to be a continuous -norm if for each .

Note that we put whenever .

Definition 2.13 (see [46]). A continuous -norm on is said to be continuous -representable if there exist a continuous -norm and a continuous -conorm on such that, for all ,

For example, for all are continuous –representable.

Define the mapping from to by:

A negation on is any decreasing mapping satisfying and . If for all , then is called an involutive negation. In the following is endowed with a (fixed) negation .

#### 3. Distribution Functions and Fuzzy Sets

We denote, , the space of all distribution functions, that is, the space of all mappings , such that is left-continuous and nondecreasing on , and . is a subset of consisting of all functions for which , where denotes the left limit of the function at the point , that is, . The space is partially ordered by the usual point-wise ordering of functions, that is, if and only if for all in . The maximal element for in this order is the distribution function given by

Example 3.1. The function , defined by is a distribution function. Since , then . Note that for each .

Example 3.2. The function , defined by is a distribution function. Since , then . Note that for each .

Example 3.3 (see [6]). The function , , defined by is a distribution function. Since , then . Note that for each .

Definition 3.4. If is a collection of objects denoted generically by , then a fuzzy set in is a set of ordered pairs: is called the membership function or grade of membership of in which maps to the membership space . Note that, when contains only the two points 0 and 1, is nonfuzzy and is identical to the characteristic function of a non-fuzzy set. The range of the membership function is or a complete lattice.

Example 3.5. Consider the fuzzy set which is real numbers considerably larger than 10, where

Example 3.6. Consider the fuzzy set which is real numbers close to 10, like (3.6) where
Note that, in this paper, in short, we apply membership functions instead of fuzzy sets.

Definition 3.7 (see [47]). Let be a complete lattice and let be a nonempty set called the universe. An - fuzzy set in is defined as a mapping . For each in , represents the degree (in ) to which is an element of .

Lemma 3.8 (see [46]). Consider the set and operation defined by for all . Then is a complete lattice.

Definition 3.9 (see [48]). An intuitionistic fuzzy set in the universe is an object , where and for all are called the membership degree and the nonmembership degree, respectively, of in and, furthermore, satisfy .

Example 3.10. Consider the intuitionistic fuzzy set which is real numbers considerably larger than 10 for first place and real numbers close to 10 in second place, where
As we said above, we apply .

#### 4. Random Normed Spaces

Random (probabilistic) normed spaces were introduced by Šerstnev in 1962 [1] by means of a definition that was closely modelled on the theory of (classical) normed spaces and used to study the problem of best approximation in statistics. In the sequel, we will adopt usual terminology, notation, and conventions of the theory of random normed spaces, as in [4, 68].

Definition 4.1. A Menger probabilistic metric space (or random metric spaces) is a triple , where is a nonempty set, is a continuous -norm, and is a mapping from into such that, if denotes the value of at the pair , the following conditions hold:
(PM1) for all if and only if ;
(PM2) ;
(PM3) for all and .

Definition 4.2 (see [1]). A random normed space (briefly, RN-space) is a triple , where is a vector space, is a continuous -norm, and is a mapping from into such that the following conditions hold:
(RN1) for all if and only if ;
(RN2) for all , ;
(RN3) for all and .

Example 4.3. Let be a linear normed space. Define Then is a random normed space. (RN1) and (RN2) are obvious and we show (RN3). for all and . Also, is a random normed space.

Example 4.4. Let be a linear normed spaces. Define Then is a random normed space. (RN1) and (RN2) are obvious and we show (RN3). for all and . Also, is a random normed space.

Example 4.5 (see [36]). Let be a linear normed space. Define Then is an RN-space (this was essentially proved by Mušthari in [49], see also [50]). Indeed, for all for all and obviously and . Next, for every and we have

Let be a function defined on the real field into itself with the following properties:(a) for every ;(b);(c) is strictly increasing and continuous on , and .

Examples of such functions are ; , ; , .

Definition 4.6 (see [51]). A random -normed space is a triple , where is a real vector space, is a continuous –norm, and is a mapping from into such that the following conditions hold:
(-RN1) for all if and only if ;
(-RN2) for all in , and ;
(-RN3) for all and .

Example 4.7 (see [37]). An important example is the space , where is a p-normed space and (-RN1) and (-RN2) are obvious and we show (-RN3). Let , then for each we have Now, if , then we have . Now, since we have then it follows that which implies that . Hence for all and .

By a non-Archimedean field we mean a field equipped with a function (valuation) from into such that if and only if , , and for all . Clearly and for all . By the trivial valuation we mean the mapping taking everything but 0 into 1 and . Let be a vector space over a field with a non-Archimedean non-trivial valuation , that is, that there is an such that is not in .

The most important examples of non-Archimedean spaces are -adic numbers. In 1897, Hensel [52] discovered the -adic numbers as a number theoretical analogue of power series in complex analysis. Fix a prime number . For any nonzero rational number , there exists a unique integer such that , where and are integers not divisible by . Then defines a non-Archimedean norm on . The completion of with respect to the metric is denoted by , which is called the -adic number field.

A function is called a non-Archimedean norm if it satisfies the following conditions:(i) if and only if ;(ii)for any , ;(iii)the strong triangle inequality (ultrametric); namely,

Then is called a non-Archimedean normed space. Due to the fact that a sequence is a Cauchy sequence if and only if converges to zero in a non-Archimedean normed space. By a complete non-Archimedean normed space we mean one in which every Cauchy sequence is convergent.

Definition 4.8. A non-Archimedean random normed space (briefly, non-Archimedean RN-space) is a triple , where is a linear space over a non-Archimedean field , is a continuous -norm, and is a mapping from into such that the following conditions hold:
(NA-RN1) for all if and only if ;
(NA-RN2) for all , ;
(NA-RN3) for all and .
It is easy to see that, if (NA-RN3) holds then so will (RN3) .

Example 4.9. As a classical example, if is a non-Archimedean normed linear space, then the triple , where is a non-Archimedean RN-space.

Example 4.10. Let be a non-Archimedean normed linear space. Define Then is a non-Archimedean RN-space.

Definition 4.11. Let be a non-Archimedean RN-space. Let be a sequence in . Then is said to be convergent if there exists such that for all . In that case, is called the limit of the sequence .
A sequence in is called a Cauchy sequence if for each and there exists such that for all and we have .
If each Cauchy sequence is convergent, then the random normed space is said to be complete and the non-Archimedean RN-space is called a non-Archimedean random Banach space.

Remark 4.12 (see [53]). Let be a non-Archimedean RN-space, then So, the sequence is a Cauchy sequence if for each and there exists such that for all we have

#### 5. Topological Structure of Random Normed Spaces

Definition 5.1. Let be an RN-space. We define the open ball and the closed ball with center and radius , as follows:

Theorem 5.2. Let be an RN-space. Every open ball is an open set.

Proof. Let be an open ball with center and radius with respect . Let . Then . Since , there exists such that . Put . Since , there exists such that . Now for given and such that , there exists such that . Consider the open ball . We claim . Now, let . Then . Therefore, we have Thus and hence .

Different kinds of topologies can be introduced in a random normed space [4]. The -topology is introduced by a family of neighborhoods In fact, every random norm on generates a topology (-topology) on which has as a base the family of open sets of the form

Remark 5.3. Since is a local base at , the -topology is first countable.

Theorem 5.4. Every RN-space is a Hausdorff space.

Proof. Let be an RN-space. Let and be two distinct points in and . Then . Put . For each , there exists such that . Consider the open balls and . Then clearly . For, if there exists then we have which is a contradiction. Hence is a Hausdorff space.

Definition 5.5. Let be an RN-space. A subset of is said to be R-bounded if there exists and such that for all .

Theorem 5.6. Every compact subset of an RN-space is R-bounded.

Proof. Let be a compact subset of an RN-space . Fix and . Consider an open cover . Since is compact, there exist such that Let . Then and for some . Thus we have and . Now, let Then . Now, we have Taking , we have for all . Hence is R-bounded.

Remark 5.7. In an RN-space every compact set is closed and R-bounded.

Theorem 5.8 (see [4]). If is an RN-space and is a sequence such that , then almost everywhere.

Theorem 5.9. Let be an RN-space such that every Cauchy sequence in has a convergent subsequence. Then is complete.

Proof. Let be a Cauchy sequence and let be a subsequence of that converges to . We prove that . Let and such that Since is a Cauchy sequence, there is such that for all . Since , there is positive integer such that and Then, if , then we have Therefore, and hence is complete.

Lemma 5.10. Let be an RN-space. If one defines then is a random (probabilistic) metric on , which is called the random (probabilistic) metric induced by the random norm .

Lemma 5.11. A random (probabilistic) metric which is induced by a random norm on a RN-space has the following properties for all and every scalar :
(i),
(ii).

Proof. We have Also, we have

Lemma 5.12. If is an RN-space, then
(i) the function is continuous,
(ii) the function is continuous.

Proof. If and , then as This proves (i).
Now, if , and , then as and this proves (ii).

Definition 5.13. The RN-space is said to be a random Banach space whenever is complete with respect to the random metric induced by random norm.

Lemma 5.14. Let be an RN-space and define by for each and . Then we have
(i) for every and ;
(ii) if satisfies (2.15), then for any , there is such that for any ;
(iii) the sequence is convergent with respect to a random norm if and only if . Also the sequence is a Cauchy sequence with respect to a random norm if and only if it is a Cauchy sequence with .

Proof. For (i), we find For (ii), by (2.15), for every we can find such that Thus we have for every , which implies that Since is arbitrary, we have
For (iii), note that since is continuous, is not an element of the set as soon as . Hence we have for every . This completes the proof.

Definition 5.15. A function from an RN-space to an RN-space is said to be uniformly continuous if for given and , there exist and such that implies

Theorem 5.16 (uniform continuity theorem). If is continuous function from a compact RN-space to an RN-space , then is uniformly continuous.

Proof. Let and be given. Then we can find such that Since is continuous, for each , we can find and such that implies But and then we can find such that Since is compact and is an open covering of , there exist in such that Put and , . For any , if then Since , there exists such that Hence we have Now, since we have it follows that Now, we have Hence is uniformly continuous.

Remark 5.17. Let be a uniformly continuous function from RN-space to RN-space . If is a Cauchy sequence in , then is also a Cauchy sequence in .

Theorem 5.18. Every compact RN-space is separable.

Proof. Let be the given compact RN-space. Let and . Since is compact, there exist in such that In particular, for each , we can choose a finite subset such that in which . Let Then is countable. We claim that . Let . Then for each , there exists such that . Thus converges to . But since for all , . Hence is dense in and thus is separable.

Definition 5.19. Let be any nonempty set and be an RN-space. Then a sequence of functions from to is said to be uniformly convergent to a function from to if for given and , there exists such that for all and .

Definition 5.20. A family of functions from an RN-space to a complete RN-space is said to be equicontinuous if for any and , there exist and such that for all .

Lemma 5.21. Let be an equicontinuous sequence of functions from an RN-space to a complete RN-space . If converges for each point of a dense subset of , then converges for each point of and the limit function is continuous.

Proof. Let and be given. Then we can find such that Since is equicontinuous family, for given and , there exist and such that, for each , for all . Since is dense in , there exists and converges for that . Since is a Cauchy sequence, for any