Filters and Ultrafilters as Approximate Solutions in the Attainability Problems with Constrains of Asymptotic Character

Problems about attainability in topological spaces are considered. Some nonsequential version of the Warga approximate solutions is investigated: we use filters and ultrafilters of measurable spaces. Attraction sets are constructed.


Introduction
This investigation is devoted to questions connected with attainability under constraints; these constraints can be perturbed. Under these perturbations, jumps of the attained quality can arise. If perturbation is reduced to a weakening of the initial standard constraints, then we obtain some payoff in a result. Therefore, behavior limiting with respect to the validity of constraints can be very interesting. But, the investigation of possibilities of the above-mentioned behavior is difficult. The corresponding "straight" methods are connected with constructions of asymptotic analysis. Very fruitful approach is connected with the extension of the corresponding problem. For example, in theory of control can be used different variants of generalized controls formalizable in the corresponding class of measures very often. In this connection, we note the known investigations of J. Warga (see [1]). We recall the notions of precise, generalized, and approximate controls (see [1]). In connection with this approach, we recall the investigations of R.V. Gamkrelidze [2]. For problems of impulse control, we note the original approach of N.N. Krasovskii (see [3]) connected with the employment of distributions. If is useful to recall some asymptotic constructions in mathematical programming (see [4,5]). We note remarks in [4,5] connected with the possible employment of nonsequential approximate (in the Warga terminology) solutions-nets.
The above-mentioned (and many other) investigations concern extremal problems. But, very important analogs are known for different quality problems. We recall the fundamental theorem about an alternative in differential games established by N.N. Krasovskii and A.I. Subbotin [6]. In the corresponding constructions, elements of extensions are used very active. Moreover, approximate motions were used. The concrete connection of generalized and approximate elements of the corresponding constructions was realized by the rule of the extremal displacement of N.N. Krasovskii. In general, the problem of the combination of generalized and approximate elements in problems with constraints is very important. Namely, generalized elements (in particular, generalized controls) can be used for the representation of objects arising by the limit passage in the class of approximate elements (approximate solutions). These limit objects can be consider as attraction elements. Very often these elements suppose a sequential realization (see [1,ch. III,IV]). But, in other cases attraction elements should be defined by more general procedures.
So, we can consider variants of generalized representation of asymptotic objects. This approach is developed by J. Warga in theory of control.
Similar problems can arise in distinct sections of mathematics. For example, adherent points of the filter base in topological space can be considered as attraction elements. Of course, here nonsequential variants of the limit passage are required very often.
In the following, the attainability problem with constraints of asymptotic character is considered.
Fix two nonempty sets E and H, and an operator h from E into H. Elements of E are considered as solutions (sometimes controls) and elements of H play the role of estimates. We consider h as the aim mapping. If we have the set E o , E o ⊂ E, of admissible (in traditional sense) solutions, then h 1 (E o ) = {h(x) : x ∈ E o } play the role of an attainability domain in the estimate space. But, we can use another constraints: instead of E o , a nonempty family E of subsets of E is given. In this case, we can use sequences (x i ) ∞ i=1 in E with a special property in the capacity of approximate solutions. Namely, we require that the sequence (x i ) ∞ i=1 has the following property: for any E o ∈ E, the inclusion x j ∈ E o takes place from a certain index (i. e. for j j o , where j o is a fixed index depending on E o ). For such solutions we obtain the sequences h(x i ) ∞ i=1 in H. If H is equipped with a topology t, then we can consider the limits of such sequences h(x i ) ∞ i=1 as attraction elements (AE) in (H, t). Of course, our AE are "sequential": we use the limit passage in the class of sequences. This approach can be very limiting. The last statement is connected both with our family E and with topology t. The corresponding examples are known: see [7,8]. In many cases, the more general variants of the limit passage are required. Of course, we can consider nets (x α ) in E and, as a corollary, the corresponding nets h(x α ) in H. In addition, the basic requirement of admissibility it should be preserved: for any E o ∈ E, the inclusion x α ∈ E o is valid starting from a certain index. With the employment of such nets, we can realize new AE; this effect takes place in many examples.
But, the representation of the "totality" of above-mentioned (Eadmissible) nets as a set is connected with difficulties. Really, any net in the set E is defined by a mapping from a nonempty directed set (DS) D into E. The concrete choice of D is arbitrary (D is a nonempty set). Therefore we have the very large "totality" of nets with the point of view of traditional Zermelo axiomatics. But, this situation can be corrected by the employment of filters of E : it is possible to introduce the set of all E-admissible filters of the set E. In addition, the E-admissibility of a filter F is defined by the requirement E ⊂ F. So, we can consider nonsequential approximate solutions (analogs of sequential approximate solutions of Warga) as filters F of E with the property E ⊂ F. Moreover, we can be restricted to the employment of only ultrafilters (maximal filters) with the abovementioned property. In two last cases, we obtain two variants of the set of admissible nonsequential approximate solutions defined in correspondence with Zermelo axiomatics. In our investigation, such point of view is postulated. And what is more, we give the basic attention to the consideration of ultrafilters. Here, the important property of a compactness arises. Namely, the corresponding space of ultrafilters is equipped with a compact topology. This permits to consider ultrafilters as generalized elements (GE) too (we keep in mind the above-mentioned classification of Warga).
The basic difficulty is connected with a realizability: the existence of free ultrafilters (for which effects of an extension are realized) is established only with the employment of axiom of choice. Roughly speaking, free ultrafilters are "invisible". This property is connected with ultrafilters of the family of all subsets of the corresponding "unit". But, we can to consider ultrafilters of measurable spaces with algebras and semialgebras of sets. We note that some measurable spaces admitting the representation of all such ultrafilters are known (see, for example, [9, § 7.6]; in addition, the unessential transformation with the employment of finitely additive (0,1)-measures is used).

General notions and designations
We use the standard set-theoretical symbolics including quantors and propositional connectives; as usually ∃! replaces the expression "there exists and unique", △ = is the equality by definition. In the following, for any two objects x and y, {x; y} is the unordered pair of x and y (see [10,ch. II]). Then, {x} △ = {x; x} is singleton containing an object x. Of course, for any objects x and y (x, y) △ = {x}; {x; y} is the ordered pair of objects x and y; here, we follow to [10,ch. II]. By ∅ we denote the empty set. By a family we call a set all elements of which are sets.
By P(X) we denote the family of all subsets of a set X; then, P ′ (X) △ = P(X) \ {∅} is the family of all nonempty subsets of X. Of course, for any set A, in the form of P ′ P(A) and P ′ P ′ (A) , we have the family of all nonempty subfamilies of P(A) and P ′ (A) respectively.
If X is a set, then we denote by Fin(X) the family of all finite sets of P ′ (X); then (FIN)[X] △ = Fin(X) ∪ {∅} is the family of all finite subsets of X.
For any sets A and B, we denote by B A the set of all mappings from In the following, N △ = {1; 2; . . .} and R is the real line; N ⊂ R. Of course, we use the natural order of R. If n ∈ N, then Transformations of families. For any nonempty family A and a set B, we suppose that If X and Y are sets and f ∈ Y X , then we suppose that (we keep in mind that P(E) is a nonempty set (∅ ∈ P(E)) and, for R ∈ P(E), R is a family) and (of course, for H ∈ P ′ (E), H is a nonempty family); moreover So, for any nonempty family E, we obtain that Special families. Let I be a set. Then, we suppose that elements of (2.2) are called π-systems with "zero" and "unit". Moreover, elements of (2.3) are lattices of subsets of I (with "zero"). Finally, Of course, in (2.4) lattices of sets with "zero" and "unit" are introduced. We note that is the set of all topologies of I. If τ ∈ (top)[I], then the pair (I, τ ) is a topological space (TS); (2.6) in (2.6) we have families dual with respect to topologies. It is obvious that (2.7) We suppose that C I : P ′ P(I) → P ′ P(I) is the mapping for which From (2.5) -(2.8), we obtain the following properties: Of course, in (2.9), we have (in particular) the natural duality used in general topology. Let (the set of all compact topologies of I). Now, we introduce in consideration algebras of sets. Namely, (2.10) In connection with (2.10), we note that If L ∈ (alg)[I], then (I, L) is a measurable space with an algebra of sets.
If L ∈ π[I], n ∈ N and A ∈ P(I), then by ∆ n (A, L) we denote the set of all mappings (L i ) i∈1,n : 1, n −→ L for each of which: is the set of all semialgebras of subsets of I. Of course, see (2.10). If we have a semialgebra of subsets of I, then algebra generated by the initial semialgebra is realized very simply: for any Now, we introduce some notions important for constructions of general topology. Namely, we consider topological bases of two types: Moreover, the following obvious property is valid: We note the natural connection of open and closed bases: (2.14) Along with (2.14), we note the following important property: (2.15) From (2.9) and (2.15), we obtain the obvious statement: So, closed bases can be used (see (2.16)) for topologies constructing. We note the following obvious property (here we use (2.14) and (2.16)):

Nets and filters as approximate solutions under constraints of asymptotic character
In this section, we fix a nonempty set E considered (in particular) as the space of usual solutions. We consider families E ∈ P ′ P(E) as constraints of asymptotic character. Of course, in this case, we use asymptotic version of solutions. The simplest variant is realized by the employment of sequences in E : in the set E N , the set of E-admissible sequences (see section 1) is selected. It is logical to generalize this approach: we keep in mind the employment of nets. Later, we introduce some definitions connected with the Moore-Smith convergence. But, before we consider the filter convergence.
We denote by β[E] (by β o [E]) the set of all families B ∈ P ′ P(E) (families B ∈ P ′ P ′ (E) ) for which is the set of all filter bases on E. By F[E] we denote the set of all filters on E : Using (3.1), we introduce the set F u [E] of all ultrafilters on E : In connection with ( and N τ (x) in correspondence with (3.3). We were introduce the filter of neighborhoods of x in the sense of [11, ch. I]. In the following, So, we introduce the closure operation in a TS. Moreover, we suppose that (3.4) The filter convergence. We follow to [11, ch. I] (3.5) In addition, (3.1). Therefore, we can use (3.5) in the Of course, it is possible to use the variant of (3.6) corresponding to Nets and the Moore-Smith convergence. On the basis of (3.6), we can to introduce the standard Moore-Smith convergence of nets. We call a net in the set E arbitrary triplet (D, , f ), where (D, ) is a nonempty DS and f ∈ E D . If (D, , f ) is a net in the set E, then (3.7) we obtain the filter of E associated with (D, , f ). Now, for any topology τ ∈ (top)[E], a net (D, , f ) in the set E, and x ∈ E, we suppose that From (3.6) and (3.7), we obtain that (3.8) is the "usual" Moore-Smith convergence (see [12, ch. 2]). Of course, any sequence x In this case, E can be considered as a constraint of asymptotic character and (D, , f ) plays the role of nonsequential (generally speaking) approximate solution.
In conclusion, we note that In (3.9), trivial ultrafilters are defined.

Attraction sets
In this section, we construct nonsequential (generally speaking) attraction sets (AS) using different variants of the representation of approximate solutions. Since nets are similar to sequences very essential, we begin our consideration with the representation (of AS) using nets. For a brevity, in this section we fix following two nonempty sets: X and Y. In addition, under f ∈ Y X and B ∈ β o [X], (4.1) of course, in (4.1), we can use a filter or ultrafilter instead of B. In addition, the important property takes place: if f ∈ Y X and B ∈ β o [X], then So, by (4.2) image of an ultrafilter base is an ultrafilter base. Of course, the image of an ultrafilter is an ultrafilter base also.
and X ∈ P ′ P(X) , then by (as)[X; Y ; τ ; f ; X ] we denote the set of all y ∈ Y for each of which there exists a net (D, , g) in the set X such that we consider (as)[X; Y ; τ ; f ; X ] as AS. In this definition, we use nets. But, for any filter F ∈ F[X] there exists a net (D, , g) in the set X for which F = (X − ass)[D; ; g] (see [13, § 1.6]).
, and X ∈ P ′ P(X) . Suppose that A and B are the sets on the left and right sides of (4.4) respectively. Let y * ∈ A. Then y * ∈ Y and, for a net (D, , g) in X, the relation  Let H * ∈ N τ (y * ). Then by (3.7) and (4.6), for some d * ∈ D, the following property is valid: In addition, D * △ = {d ∈ D | d * d} ∈ P ′ (D) and by (3.7) and (4.5) As a corollary, and, as a corollary, by (3.5) and (4.9) By (3.7) and the choice of (D, ⊑ , ϕ), for some d o ∈ D, the following property is realized: By the choice of F o we obtain that ∀δ ∈ D . For any f ∈ Y X , τ ∈ (top) [Y ], and X ∈ P ′ P(X) Proof. We denote respectively by F an U the sets on the left and right sides of (4.12).
and, as a corollary, . By the choice of F we have the inclusion (see (3.5)). Then by (4.13 and, as a corollary (see (3.5)), Recall that, for any family X ∈ P ′ P(X) , {∩} f (X ) ∈ P ′ P(X) and X ⊂ {∩} f (X ). We note the following obvious On the other hand, from (3.1), we obtain that and, as a corollary, , and X ∈ P ′ P(X) , then The corresponding proof is realized by the immediate combination of Propositions 4.2 and 4.3. We note that, by definitions of Section 2 (4.14) In connection with (4.14), we note the following general property.
Then, by (4.14), (4.15), and Corollary 4.1 (4. 16) In connection with (4.16), we note that ∀X ∈ P ′ P(X) Remark 4.1. By analogy with Proposition 4.3 we have that Returning to (4.17), we note that by Proposition 4.2 ∀f ∈ Y X ∀X ∈ P ′ P(X) Indeed, consider the case X = Y = R, f (x) = x ∀x ∈ X, τ = τ R is the usual | · |-topology of real line R, and Then, It is obvious the following Proof. The corresponding proof follows from known statements of general topology (see [11, ch. I]). But, we consider this proof for a completeness of the account. In our case, we have (4.15). In addition, is nonempty family of sets closed in the compact topological space (we use known properties of the closure operation and the image operation). Since ∅ / ∈ B, we obtain that ∅ / ∈ T . In addition, T ∈ β[Y ]. Therefore, by [9, (3.3.16)] we have the following property: if n ∈ N and As a corollary, T is the nonempty centered system of closed sets in a compact TS. Then, the intersection of all sets of T is not empty. By (4.19) Using (4.15), we obtain the required statement about the nonemptyness of attraction set.
In the following, we use the continuity notion. In this connection, suppose that (4.20) So, continuous functions are defined. In the following, we use bijections, open and closed mappings, and homeomorphisms. Let In (4.22) (in (4.23)), we consider open (closed) mappings. In addition, (4.24) So, in (4.24), the set of homeomorphisms is defined.

Some properties of ultrafilters of measurable spaces
In this section, we fix a nonempty set E. We consider the very general measurable space (E, L), where L ∈ π[E] is fixed also. According to necessity, we will be supplement the corresponding suppositions with respect to L. We suppose that F * (L) is the set of all families F ∈ P ′ (L) such that Elements of the set F * (L) are filters of L. In addition, is the set of all ultrafilters of L. Recall that (see [16, p. 29]) In the following, (5.2) plays the very important role. We introduce the mapping Φ L : L → P F * o (L) by the following rule: We note that {E} ∈ F * (L) and by (5 In addition, we recall that (see Section 2) is a nonempty multiplicative space. We note some simplest general properties. We obtain that With the employment of (5.5), we obtain that, for any Now, we return to the space (E, L). Suppose that We note the obvious property: . In addition,

Using (5.5) and the obvious inclusion
has the following properties . We can use this property in (5.8): for any F ∈ F * (L) and In connection with (5.9), we recall the very general property: if Using the maximality property, we obtain that Of course, the above-mentioned properties are valid for (5.10) The following reasoning is similar to the construction of [13, § 3.6] connected with Wallman extension; in addition, later until the end of this section, we suppose that (5.10) is valid (so, we fix a lattice with "zero" and "unit"). So, if U ∈ F * o (L), A ∈ L, and B ∈ L, then (under condition (5.10)) The property (5.11) is basic. As a corollary, We note that by (5.11) the following property is valid: As a corollary, we obtain the property 6 Topological properties, 1 As in the previous section, now we fix a nonempty set E and a family L ∈ π[E]. We note the following obvious property: From definitions of the previous section, the following known property follows: As a corollary, we obtain (see Section 2) that We recall the very known definition of Hausdorff topology; namely, we introduce the set of such topologies: if M is set, then For any set M we suppose that , then TS (M, τ ) is called a compactum. Then, the obvious statement follows from the ultrafilter properties (see (5.3), (6.1)): ) is a Hausdorff TS. Of course, we can use the previous statements of this section in the case of L ∈ (LAT) o [E], obtaining the Hausdorff topology (6.5). But, in the above-mentioned case, another construction of TS is very interesting. This construction is similar to Wallman extension (see [13, § 3.6]). Moreover, in this connection, we note the fundamental investigation [14], where topological representations in the class of ideals are considered. We give the basic attention to the filter consideration in connection with construction of Section 3 concerning with the realization of AS. In this connection, we note that P(E) ∈ π[E] and the sets F * P(E) and F * o P(E) are defined. From (3.1) and definitions of Section 5, we have the equality F * P(E) = F[E]. Moreover, from (3.2) and the above-mentioned definitions of Section 5, the equality follows. By these properties (see (6.6)) the constructions of Section 3 obtain interpretation in terms of filters and ultrafilters of measurable spaces. Now, we note one simple property; in addition, we use the inclusion In particular, we have the following property: We note one general simple property; namely, in general case of L ∈ Remark 6.1. We note that (6.8) is a variant of Proposition 2.4.1 of monograph [16]. Consider the corresponding proof. Fix U ∈ F * o (L). Then by (6.7) Then, W ∈ F u [E] and V ⊂ W. In addition (see Section 5), W ∩ L ∈ F * (L). Let U ∈ U . Then, U ∈ L and, in particular, U ∈ P(E). By (6.9) U ∈ V and, as a corollary, U ∈ W. Then, U ∈ W ∩ L. So, the inclusion U ⊂ W ∩ L is established; we obtain that W ∩ L ∈ F * (L) : U ⊂ W ∩ L. (6.10) From (5.1) and (6.10), we have the equality U = W ∩ L. So, Since the choice of U was arbitrary, the property (6.8) is established.

Topological properties, 2
In this and following sections, we fix a nonempty set E and a lattice We consider the question about constructing a com- . This space is similar to Wallman extension for a T 1 -space. But, we not use axioms of topology and operate lattice constructions (here, a natural analogy with constructions of [14, ch. II] takes place). Later we use the following simple statement.
By (2.15) and Proposition 7.1 we have the following construction: Proposition 7.2. . The following compactness property is valid: Proof. For brevity, we suppose that has the following obvious property T. (7.5) We have the equality Let η be arbitrary nonempty centered subfamily of U (for any m ∈ N and (T i ) i∈1,m ∈ η m the intersection of all sets T i , i ∈ 1, m, is not empty). If H ∈ η, then the family has the property: D H ⊂ U ∀U ∈ H. Of course, is centered. Indeed, choose n ∈ N and (Λ i ) i∈1,n ∈ L n . Let ( H i ) i∈1,n ∈ η n be a procession with the property: Then, in particular, (Λ i ) i∈1,n ∈ L n . In addition, by (7 Since the intersection of all sets H i , i ∈ 1, n, is not empty (we use the centrality of η), we choose an ultrafilter Then, Λ j ∈ U under j ∈ 1, n. By axioms of a filter (see Section 5) we obtain that Since L is closed with respect to finite intersections, we obtain that Moreover, (7.8) is supplemented by the following obvious property; namely, . As a corollary, Then, the intersection of all sets of η is not empty. Since the choice of η was arbitrary, it is established that any nonempty centered family of closed (in TS (7.6)) sets has the nonempty intersection. So, TS (7.6) is compact (see [11][12][13] We have the nonempty compact TS The corresponding proof follows from (6.2); of course, we use (5.4) also. From (2.18), (7.11), and Proposition 7.3, we obtain the following property: So, by (7.13) we obtain that (7.12) is a nonempty compact T 1 -space.
In conclusion of the given section, we note several properties. First, we recall that In addition, from (7.11), the obvious representation follows: (7.15) With the employment of (7.15) the following statement is established.
is a local base of TS (7.12) at U : We note that, from definitions, the following property is valid:

The density properties
In this section, we continue the investigation of TS (7.12). Of course, we preserve the suppositions of Section 7 with respect to E and L. But, in this section, we postulate that {x} ∈ L ∀x ∈ E. So, in this section and {x} ∈ L ∀x ∈ E. Therefore, with regard (3.9) and (8.1), we obtain that . We use Proposition 7.4. Namely, we choose a set L ∈ L \ U for which Since E ∈ U by axioms of a filter (see Section 5), we obtain that L = E.
Since the choice of H was arbitrary, . Since the choice of U was arbitrary, the inclusion is established. The inverse inclusion is obvious (see (7.11)).
So, we obtain that trivial ultrafilters (8.2) realize an everywhere dense set in the TS (7.12).
Returning to (7.11), we note one obvious property connected with (7.16). Namely, by (2.14) and Proposition 7.1, in general case of and, in particular, And what is more by (2.17), (7.11), and Proposition 7.1, in general is a base of topology (7.11). We recall that by (2.8)   Until the end of the present section, we suppose that So, in our case, (E, L) is the lattice of closed sets in T 1 -space. Then, (7.12) is the corresponding Wallman compact space (see [13, ch. 3]).
On the other hand, by (8.10) and (8.11) we obtain that this variant of (E, L) corresponds to general statements of our section (for example, see (8.2) and Proposition 8.1). In this connection, we consider the mapping we denote the mapping (8.12) by f .
Consider some simple properties. First, we note that f is injective: Indeed, for x 1 ∈ E and x 2 ∈ E with the property f (x 1 ) = f (x 2 ), by (3.9) we have that {x 1 } ∈ f (x 2 ) and, as a corollary, x 2 ∈ {x 1 }; so, Of course, f is a bijection from E onto the set If L ∈ L and x ∈ E, then L ∈ f (x) ⇔ (x ∈ L). As a corollary, we obtain that As a result, we obtain that Since the choice of F was arbitrary, from (8.19) we obtain the required continuity property (see [16, (2.5.2)]).

Ultrafilters of measurable space
In this Section, we fix a nonempty set I and an algebra A of subsets of I. So, in this section, (I, A) is a measurable space with an algebra of sets: A ∈ (alg)[I]. Of course, we can to use constructions of Section 5; indeed, in particular, we have the inclusion A ∈ (LAT o )[I]; see (2.10). As a corollary, by (2.4) A ∈ π[I]. So, we use the sets F * (A) and F * o (A) of Section 5; we use properties of these sets also. We note the known representation (see [15, ch. I]): Now, we use (9.1) for investigation of TS (7.12) in the case L = A. First, we note the obvious corollary of (9.1): . Then by (7.14) A / ∈ U 1 . By (9.1) I \ A ∈ U 1 , where I \ A ∈ A by axioms of an algebra of sets. So, by ( is established. Let U 2 ∈ Φ A (I \ A). Then, by (5.3) U 2 ∈ F * o (A) and I \ A ∈ U 2 . By axioms of a filter So, A / ∈ U 2 and U 2 / ∈ Φ A (A). As a corollary, Returning to (9.2) in general case, we note the following obvious  . Therefore, we obtain the following property: Since the choice of Λ (9.5) was arbitrary, the inclusion So, for measurable spaces with algebras of sets, the topological representations of Sections 6 and 7, 8 realize the same topology. By (6.5), (7.13), and Proposition 9.2 So, we obtain a nonempty compactum. Recall that (see (7.11), Proposition 9.
Since Ω is open, then by (6.4) we obtain that, for some family W ∈ P (UF)[I; A] , (9.14) the following equality is realized: With the employment of (9.1), the following obvious property is established: in our case of measurable space with an algebra of sets 3 For a completeness, we consider the scheme of the proof of (9.21). For this, we note that by (3.9)  Of course, by (3.9), for A ∈ A, we have the following obvious implications: Then, by (9.23) (A ∈ F * ) ∨ (I \ A ∈ F * ). Since the choice of A was arbitrary, by (9.1) F * ∈ F * o (A). So, (9.21) is established. Using (9.21), we introduce the mapping Of course, in (9.24) we have analog of the mapping f (8.12). But, in the given case, we realize the immersion of points of the initial set in the ultrafilter space under other conditions. We will use the specific character of measurable space with an algebra of sets. Now, we note the obvious property: (9.25) In (9.25), the statement of the premise has the following sense: algebra A is distinguishing for points of I. If J ∈ P ′ (A), then by analogy with Section 4 we suppose that and moreover the following property is valid: We note that Proposition 9.4 is similar to Proposition 8.1. But, in the given section, the condition {x} ∈ A ∀x ∈ A (9.34) was supposed not. In construtions of Section 8 (in particular, in Proposition 8.1), the condition similar to (9.34) is essential. So, Proposition 9.4 has the independent meaning.

Attraction sets under the restriction in the form of algebra of sets
In the following, we fix a nonempty set E, a TS (H, τ ), where H = ∅, and a mapping h ∈ H E . Elements e ∈ E are considered as usual solutions and elements y ∈ H play the role of some estimates. The natural variant of an obtaining of y is realized in the form y = h(e), where e ∈ E. But, we admit the possibility of the limit realization of y. This is natural in questions of asymptotic analysis. In the last case, it is natural to use "asymptotic constraints" in the form of a nonempty subfamilies of P(E). Then, we obtain constructions of Section 4 under X = E, Y = H, and f = h. But, we admit yet one possibility: along with "usual" AS, we use the sets Of course, we use remarks of the conclusion of the previous section. (see (3.5)). In addition, by the choice of F we have the inclusion E ⊂ F; see (9.26). By (9.27), for some U ∈ F * o (A| E), the inclusion F ⊂ U is valid. Then, Section 9). Then, by (3.5)
So, by (10.1) and (10.2) some "partial" AS are defined. Of course, the case for which (10.10) is converted in a equality is very interesting. For investigation of this case, we consider auxiliary constructions. In the following, in this section, we fix A ∈ (alg)[E]. So, (E, A) is a measurable space with an algebra of sets. In this case, we can supplement the property (6.8). Namely, Remark 10.1. We omit the sufficiently simple proof of (10.15). Now, we are restricted to brief remarks. Namely, by ultrafilter U ∈ F u [E] we can realize a finitely additive (0,1)-measure µ on the family P(E) supposing that µ(L) △ = 1 under L ∈ U and µ(Λ) △ = 0 under Λ ∈ P(E) \ U . In connection with such possibility, we use [9, (7.6.17)] (moreover, see [9, (7.6.7)]). The natural narrowing ν of µ on our algebra A is finitely additive (0,1)-measure on A (of course, ν = (µ| A)). Therefore, for some V ∈ F * o (A), by [9, (7.6.17)] ν is defined by the rule On the other hand, the family U ∩ A realizes ν by the obvious rule: Therefore, for any G * ∈ N o τ (z) there exists B * ∈ B such that h 1 (B * ) ⊂ G * . As a corollary, . Since the choice of G * was arbitrary, Choose arbitrary neighborhood G * ∈ N o τ (z). Then, by (10.22) and by (10.20 The proof of (10.21) is completed. We note that, in (10.21), we can use instead of B arbitrary filter of (E, A). In this connection, we recall that by constructions of Section 5, for any F ∈ F * (A), we obtain (in particular) that F ∈ β o [E] and Now, with the employment of (10.26), we obtain (10.25). We note the following obvious corollary of (10.25) (in this connection, we recall (10.21)): follows. Since the choice of G was arbitrary, the inclusion Using the last implication and (10.28), we obtain the required property (10.27). Using (10.15), we obtain the obvious corollary of (10.27): Until the end of the present section, we suppose that Condition 10.1 is valid.  Section 9) and Recall that E ⊂ U . Since E ∈ P ′ (A), we obtain that E ⊂ U ∩ A. Therefore (see (9.26)),

Ultrasolutions
First, we recall some statements of [17]. In addition, we fix a nonempty set E and a TS (H, τ ), where H = ∅. We consider the nonempty set F u [E]. Suppose that h ∈ H E . Then, we suppose that (11.1) So, we introduce the limit sets corresponding to ultrafilters of E. By analogy with Proposition 5.4 of [17] the following statement is established.
The obtained contradiction means that y 1 = y 2 is impossible. So, y 1 = y 2 .
Proof. The corresponding proof is the obvious combination of (11.1), (11.5) The uniqueness of y is obvious.
, then the following equality is valid: Proof. Let u △ = H[τ ](U ). Then u ∈ H and by (11.16) Then, by (3.5 . Therefore, for any T ∈ N τ (u), there exists U ∈ U such that h 1 (U ) ⊂ T (see (3.3) and (4.1)). Let A * ∈ U . Then, h 1 (A * ) ∈ P(H). If S ∈ N τ (u), then, for some U S ∈ U , the inclusion h 1 (U S ) ⊂ S is valid; moreover, A * ∩ U S = ∅ and Since the choice of S was arbitrary, we obtain that Therefore, u ∈ cl h 1 (A * ), τ . Since the choice of A * was arbitrary too, we have the inclusion u ∈ Then, for W ∈ N τ (q), we obtain the property W ∩ T = ∅ ∀T ∈ h 1 [U ]. Using (3.3), we have the following statement: By (11.5) u = q. Then q ∈ {u}. Since the choice of q was arbitrary, we obtain that The opposite inclusion was established previously. Therefore, {u} and the intersection of all sets cl h 1 (A), τ , A ∈ U , coincide.
are fulfilled. Therefore, by (3.6) the following two properties are valid: such that T ⊂ S; then, T ∈ U by (12.2) and S ∈ U by axioms of a filter). In addition, Z ⊂ N τ (z). Then, Since Z ∈ (z − bas)[τ ], by (10.27) We note that (see [11][12][13]) by (11.11)   In connection with Proposition 12.5, we recall Proposition 9.2 and known statement about the possibility of an extension of continuous functions defined on the initial space; in this connection, see, for example, Theorem 3.6.21 of monograph [13]. For this approach, constructions of Section 8 are essential. Of course, under corresponding conditions, we can use the natural connection with the Wallman extension (see (8.27) and Proposition 9.2).
In this case, Proposition 12.5 can be "replaced" (in some sense) by statements similar to the above-mentioned Theorem 3.6.21 of [13] (of course, this approach requires a correction, since we consider ultrafilters of the measurable space). But, we use the "more straight" way with point of view of asymptotic analysis: we construct the required continuous mapping by the limit passage (see Proposition 12.5). We recall (9.24). Then, by (9.24) and (12.5) the mapping

Ultrafilters as generalized solutions
We suppose that E, (H, τ ), h, and A satisfy to the conditions of Section 12. We postulate (11.11). Finally, we postulate Condition 10.1. Therefore, we can use constructions of the previous section. In particular, (12.25) is fulfilled (the more general property (12.24) is fulfilled too). In connection with (12.25), the obtaining of more simple representations of AS is important. For this goal, we use the natural construction of Theorem 8.1 in [17]. Namely, we have the following Of course, A ∈ P(E). And what is more, A ∈ F. Indeed, let us assume the contrary: A ∈ E \ F. (13.4) Recall that E ⊂ A. Therefore, A ∈ A. By (9.1) and ( We have the obvious contradiction. This contradiction means that (13.4) is impossible. So, A ∈ F. Since the choice of A was arbitrary, the inclusion E ⊂ F is established. Then (see (9.26)), F ∈ F * o (A| E). So, we obtain the inclusion is established. So, by (13.9) and (13.14) we obtain that the net (D, ⊑ , g) in the set E has the following properties: We note that, in Theorem 13.1, the set F * o (A| E) plays the role of the set of admissible generalized solutions.

Some remarks
In our investigation, one approach to the representation of AS and approximate solutions is considered. This very general approach requires the employment of constructions of nonsequential asymptotic analysis. This is connected both with the necessity of validity of "asymptotic constraints" and with the general type of the convergence in TS. We fix a nonempty set of usual solutions (the solution space), the estimate space, and an operator from the solution space into the estimate space. In the estimate space, a topology is given. Then, under very different constraints, we can realize in this space both usual attainable elements and AE. But, if usual attainable elements are defined comparatively simply (in the logical relation), then AE are constructed very difficult. For last goal, extensions of the initial space are used. In addition, the corresponding spaces of GE are constructed. Ultrafilters of the initial space can be used as GE. But, the realizability problem arise: free ultrafilters are "invisible". In addition, free ultrafilters realize limit attainable elements which nonrealizable in the usual sense. In this connection, we propose to use ultrafilters of (nonstandard) measurable space; we keep in mind spaces with an algebra of sets. But, it is possible to consider the more general constructions with the employment of ultrafilters. In our investigation, ultrafilters of lattices of sets are used. On this basis, the interesting connection with the Wallman extension in general topology arises.
It is possible that the proposed approach motivated by problems of asymptotic analysis can be useful in other constructions of contemporary mathematics.