A mathematical–physical approach on regularity in hit-and-miss hypertopologies for fuzzy set multifunctions

In this paper, an approach concerning hit-and-miss hypertopologies and especially regularity property viewed both as a continuity property in a hit-and-miss hypertopology (from a mathematical point of view) and also as a physical approximation property is intended.


Introduction
Nowadays, Hausdorff, Vietoris, Wijsman, Fell, Attouch-Wets, etc., hypertopologies are intensively studied due to their various applications in optimization, convex analysis, economics, image processing, sound analysis and synthesis (see Beer [7], Apreutesei [4], Hu and Papageorgiou [20], etc., concerning the Vietoris topology). Results involving the Hausdorff distance were obtained by Lorenzo and Maio [26] in melodic similarity, Lu et al. [27]-an approach to word image matching, etc. Recently, it was shown that using proximity, all hypertopologies known so far are of the type hit-and-miss, which led to the unification of all hypertopologies under one topology called the Bombay Hypertopology [29].
The idea of modeling at multiple scales the phenomena behavior has become a useful tool in pure mathematics, applied mathematics physics and so on. Fractals are multiscale objects, which often describe such phenomena better than traditional mathematical models do. That is why fractal-based techniques lie at the heart of these areas. Kunze et al. [21] and Wicks [42] developed hyperspace theories concerning the Hausdorff metric and the Vietoris topology, as a foundation for self-similarity and fractality. In fact, for many years, topological methods were used in many fields to study the chaotic nature in dynamical systems (see for instance Sharma and Nagar [40], Wang et al. [41], Goméz-Rueda et al. [18], Li [24], Liu et al. [25], Ma et al. [28], Fu and Xing [11], etc.). These phenomena seem to be collective (set-valued), emerging out of many segregated components, having collective dynamics of many units of individual systems. This arose the need of a topological study of such collective dynamics. Recent studies of dynamical systems, in engineering and physical sciences, have revealed that the underlying dynamics is set-valued (collective), and not of a normal, individual kind, as it was usually studied before.
Since in some examples of fractals (like neural networks and the circulatory system), the uniform property of the Hausdorff topology is inappropriate, we could intend to choose a convenient topology on the set of values of the studied multifunctions. In this sense, Wijsman topology may be preferred instead of Hausdorff topology because Wijsman topology could describe better the pointwise properties of fractals.
On the other hand, recently, domain theory has been studied in theoretical computer science, as a mathematical theory of semantics of programming languages (Edalat [10], Gierz et al. [17], etc.). In this context, (hyper)topological notions from Mathematical Analysis as well as measure theory, dynamical systems or fractality can be considered via domain theory, obtaining computational models. Namely, in denotational semantics and domain theory, power domains are domains of nondeterministic and concurrent computations. As it is well-known, domain theory was introduced by Scott in theoretical computer science as a mathematical theory of semantics of programming languages.
Together with the increasing interest in hypertopologies, non-additive set multifunctions theories developed. In this context, regularity is known as an important continuity property with respect to different topologies, but, at the same time, it can be interpreted as an approximation property. Using regularity, we can approximate ''unknown'' sets by other sets which we have more informations. Usually, from a mathematical perspective, this approximation is done from the left by closed sets, or more restrictive, by compact sets and/or from the right by open sets. As a mathematical direct application of regularity, the classical Lusin's theorem concerning the existence of continuous restrictions of measurable functions is very important and useful for discussing different kinds of approximation of measurable functions defined on special topological spaces and for numerous applications in the study of convergence of sequences of Sugeno and Choquet integrable functions (see Li et al. [23] for an interesting application of Lusin theorem), in the study of the approximation properties of neural networks, as the learning ability of a neural network is closely related to its approximating capabilities. Also, regular Borel measures are important tools in studies on the Kolmogorov fractal dimension (Barnsley [6], Mandelbrot [30], etc.). Lebesgue measure is a remarkable example of a regular measure.
The paper is organized as follows: in ''Hit-and-miss hypertopologies: an overview'' and ''Regular set multifunctions'' several remarkable hit-and-miss hypertopologies and their properties are listed from a mathematical perspective and regularity of set multifunctions is introduced in a unifying way with respect to these hypertopologies. In ''Regularization by sets of functions of eapproximation-type scale. Physical correspondences with hit-and-miss topologies'' and ''Conclusions'', a physical perspective concerning regularity and fractality is provided.

Hit-and-miss hypertopologies: an overview
Hausdorff, Vietoris and Wijsman, etc., topologies are remarkable examples of the so-called hit-and-miss hypertopologies. Like some physical concepts, these hypertopologies, although are composed of two independent parts, upper and lower hypertopologies, they become consistent when seen together. For instance, in physical terms, the non-differentiability of the curve motion of the physical object involves the simultaneous definition at any point of the curve, of two differentials (left and right). Since we cannot favor one of the two differentials, the only solution is to consider them simultaneously through a complex differential. Its application, multiplied by dt, where t is an affine parameter, to the field of space coordinates implies complex speed fields.
We now briefly recall and list the definitions and main properties of the above-mentioned hypertopologies:
Vietoris topology b s V on P 0 ðXÞ has as a subbase the class

the lower and upper Vietoris topologies:
b s þ V -the upper Vietoris topology (b s À V -the lower Vietoris topology, respectively) is the topology which has as a subbase the class S UV (S LV , respectively).
. . .; V k 2 s; is a base for the topology b s V and the family of subsets respectively). In different continuity properties (regularity for instance), the following observation is used: In what follows, let (X, d) be a metric space. By P f ðXÞ we mean the family of closed, nonvoid sets of X, by P bf ðXÞ the family of bounded, closed, nonvoid sets of X and by P k ðXÞ, the family of all nonvoid compact subsets of X Á s d denotes the topology induced by the metric d.
The following statements are equivalent: Remark 2.4 [14] (i) If (X, d) is a complete, separable metric space, then P f ðXÞ with the Wijsman topology is a Polish space (Beer [7]

Hausdorff topology
In recent years, due to the development of computational graphics (for instance, in the automatic recognition of figures problems), it was necessary to measure accurately the matching, i.e., to calculate the distance between two sets of points. This led to the need to operate with an acceptable distance, which has to satisfy the first condition in the definition of a distance: the distance is zero if and only if the overlap is perfect. An appropriate metric in these issues is the Hausdorff metric on which we will refer in the following and which, roughly speaking, measures the degree of overlap of two compact sets. Let M; N 2 P f ðXÞ: The Hausdorff-Pompeiu pseudometric h on P f ðXÞ is the ''greatest'' of all distances from any point in one of these two sets, to the nearest point from the other set, so, it is defined by ðÃÞ hðM; NÞ ¼ maxfeðM; NÞ; eðN; MÞg; where eðM; NÞ ¼ sup x2M dðx; NÞ is the excess of M over N and dðx; NÞ ¼ inf y2N dðx; yÞ is the distance from x to N (with respect to the metric d).
The topology induced by the Hausdorff pseudometric h is called the Hausdorff hypertopology s H on P f ðXÞ: On P bf ðXÞ, h becomes a veritable metric. If, in addition, X is complete, then the same is P f ðXÞ (Hu and Papageorgiou [20]). And this highlights the uniform aspect of the Hausdorff topology: it is the topology on P f ðXÞ of uniform convergence on X of the distance functionals x7 !dðx; MÞ, with M 2 P f ðXÞ: Hausdorff topology is invariant with respect to uniformly equivalent metrics (Apreutesei [4]).
In the following, we list some properties of the Hausdorff metric: If X is a Banach space, then: (II) (i) hðaM; aNÞ ¼ jajhðM; NÞ; 8a 2 R; 8M; N 2 P f ðXÞ; (ii) hðM þ P; N þ PÞ hðM; NÞ; 8M; N; P 2 P f ðXÞ; If, particularly, X ¼ R; and a; b; c; d 2 R; with a\b; c\d, then hð½a; b; ½c; dÞ ¼ maxfja À cj; jb À djg: Remark 2.6 [21] Hausdorff metric has some interesting characteristics: (i) It is possible for a sequence of finite sets to converge to an uncountable set: Barnsley [6] calls the space ðP k ðXÞ; hÞ, the life space of fractal. Recently, Banakh and Novosad [5] proposed a fractal approach using Vietoris topology (in a more general setting than the one used for the Hausdorff topology).

Regular set multifunctions
Suppose that T is a locally compact, Hausdorff space, C a ring of subsets of T and X a real normed space space. Usually, it is assumed that C is B 0 (B 0 0 , respectively)-the Baire d-ring (r-ring, respectively) generated by compact sets, which are G d (i.e., countable intersections of open sets) or C is B (B 0 , respectively)-the Borel d-ring (r-ring, respectively) generated by the compact sets of T.
Since T is locally compact, the following statements can be easily verified (Dinculeanu [9, Ch. III, p. 197]): Let l : C ! P f ðXÞ be an arbitrary set multifunction. We easily observe that l is monotone and jlðAÞj ¼ mðAÞ; for every A 2 C. (ii) Let m 1 ; . . .; m p : C ! R þ , be p finitely additive set functions, where C is a ring of subsets of an abstract space T. We consider the set multifunction l : C ! P f ðRÞ, defined for every A 2 C by lðAÞ ¼ fm 1 ðAÞ; m 2 ðAÞ; . . .; m p ðAÞg: Then the set multifunction l _ : C ! P f ðRÞ, defined for every A 2 C by: In what follows, let l : ðC; s 1 Þ ! ðP f ðXÞ; s 2 Þ be a monotone set multifunction, where s 1 2 fs;s l ;s r g and s 2 2 fs H ; s W ; s V g.
Precisely, we have:  Remark 3.7 Every K 2 K is R l -regular and every D 2 D is R r -regular.
The following results can be proved using the above definitions:   Remark 3.10 For s 2 ¼ s H ; s W or s V , respectively, we particularly get the notions of regularity as we defined and studied in [12][13][14]. For instance, if s 2 ¼ s H , then, by its monotonicity, l is (in the sense of [12]): In fact, one may easily observe that (in s H ): (i) l is regular iff for every e [ 0, there are K 2 K \ C; K & A and D 2 D \ C; D ' A so that eðlðDÞ; lðKÞÞ\e; (ii) l is R l -regular iff for every e [ 0, there is K 2 K \ C; K & A so that eðlðAÞ; lðKÞÞ\e; (iii) l is R r -regular iff for every e [ 0, there is D 2 D \ C; D ' A so that eðlðDÞ; lðAÞÞ\e; that is, in each case, we find an alternative expression of regularity as an approximation property.
Regularization by sets of functions of e-approximation-type scale: physical correspondences with hit-and-miss topologies In this section, analogously to our considerations from the previous section concerning regularity as an approximation property, we now study physical regularizations. Precisely, as we shall see, generally, the ''reduction'' of the complex dimensions to their real part requires the regularization by sets of functions of e-approximation-type scale, while the ''reduction'' to their imaginary part requires regularization with ''known'' sets, that is, sets for which we have some informations.
We consider a fractal function f(x), with x 2 ½a; b (for instance, one of the trajectory's equation) and the sequence of the variable x values: By f ðx; eÞ; we denote the fractured line connecting the points f ðx 0 Þ; . . .; f ðx k Þ; . . .; f ðx n Þ: This line will be considered as an approximation which is different from the one used before. We shall say that f ðx; eÞ is an e-approximation scale. Now, we consider the e-approximation scale f ðx; eÞ of the same function. When we study a fractal phenomenon by approximation, because f(x) is similar almost everywhere, then, if e and e are small enough, the two approximations f ðx; eÞ and f ðx; eÞ must lead to the same results. If we compare the two cases, then to an infinitesimal increase de of e, it corresponds an increase de of e, if the scale is dilated.
In this case, de e ¼ de e , i.e., is the ratio of the scale e þ de and de must be preserved. Then, we can consider the infinitesimal transformation of the scale as Finally, we get The operator is called the dilatation operator.
The above relation shows that the intrinsic variable of the resolution is not e, but ln e e 0 : On the other hand, simultaneous invariance with respect to both space-time coordinates and the resolution scale induces general scale relativity theory (SRT) [32,33]. These theories are more general than Einstein's general relativity theory, being invariant with respect to the generalized Poincaré group (standard Poincaré group and dilatation group) [32,33].
Basically, we discuss various physical theories built on manifolds of fractal space-time and they all turn out to be reducible to one of the following classes: (i) SRT [35,36] and its possible extensions [34]. It is considered that the microparticles motion takes place on continuous but non-differentiable curves. In such context, regularization works using sets of functions of e-approximation-type scale. (ii) Transition in which to each point of the motion trajectory, a transfinite set is assigned (in particular, a Cantor-type set-see the El Naschie [34] e ð1Þ model of space-time), to mimic the continuous (the trans-physics). In such context, the regularization of ''vague'' sets by known sets works. (iii) Fractal string theories containing simultaneously relativity and trans-physics [19,37].
The reduction of the complex dimensions to their real part is equivalent to Scale Relativity-Type theories, while reducing them to the imaginary part of their complex dimensions generates trans-physics. In such context, the simultaneous regularization by sets of functions of e-approximation-type scale and also by ''known'' sets works. The ''reduction'' of the complex dimensions to their real part requires the regularization by sets of functions of eapproximation-type scale, while the ''reduction'' to their imaginary part requires regularization with ''known'' sets. Dynamical systems behaviors are collective phenomena emerging out of many segregated components. Most of these systems are collective (that is, set-valued) dynamics of many units of individual systems, whence the need of a (hyper)fractal topological treatment of such collective dynamics. We consider that the particle of a complex system moves on continuous, but non-differentiable curves (fractal curves). Once accepted such a hypothesis, some consequences of non-differentiability by SRT are evident [35,36].
For instance, physical quantities that describe the complex system are fractal functions, i.e., functions depending both on spatial coordinates and time as well as on the scale resolution dt s . In classical physics, the physical quantities describing the dynamics of a complex system are continuous, but differentiable functions depending only on spatial coordinates and time.
Since [1,30,31,35,36], two representations are complementary: the formalism of the fractal hydrodynamics (at the continuum level), and the one of the Schrödinger-type theory (at the discontinuum level). Moreover, the chaoticity, either through turbulence in the fractal hydrodynamic approach, either through stochasticization in the Schrödinger-type approach, is generated only by the non-differentiability of the movement trajectories in a fractal space.

Conclusions
In this paper, we intend to present a unifying mathematical-physical perspective concerning the relationships, interpretations and similitudes existing among fractality, regularity and several hit-and-miss hypertopologies. We intend to continue the study of regularity in hypertopologies viewed in the context of domain theory (in correlation with [10,17]). We are also interested in developing a neural network fractal theory using Wijsman topology (its pointwise character seems to characterize some properties better than the Hausdorff topology induced by the Hausdorff-Pompeiu metric (which has a uniform character).