Filters and Ultrafilters as Approximate Solutions in the Attainability Problems with Constraints of Asymptotic Character ()
Abstract 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. AMS (MOS) subject classification. 46A, 49 K 40.
1. 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 abovementioned 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 realizetion (see [1]). But, in other cases, attraction elements should be defined by more general procedures.
So, we can consider variants of generalized representtation 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 
and H, and an operator h from 
into H. Elements of 
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 
of admissible (in traditional sense) solutions, then 
play the role of an attainability domain in the estimate space. But, we can use another constraint: instead of 
a nonempty family 
of subsets of E is given. In this case, we can use sequences 
in 
with a special property in the capacity of approximate solutions. Namely, we require that the sequence 
has the following property: for any 
the inclusion 
takes place from a certain index (i.e. for 
where 
is a fixed index depending on
). For such solutions we obtain the sequences 
in H. If H is equipped with a topology t, then we can consider the limits of such sequences 
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 
and with topology 
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 
in 
and, as a corollary, the corresponding nets 
in H. In addition, the basic requirement of admissibility it should be preserved: for any 
the inclusion 
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 (
-admissible) nets as a set is connected with difficulties. Really, any net in the set 
is defined by a mapping from a nonempty directed set (DS) 
into 
the concrete choice of 
is arbitrary (
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 
it is possible to introduce the set of all 
-admissible filters of the set 
In addition, the 
-admissibility of a filter 
is defined by the requirement 
So, we can consider nonsequential approximate solutions (analogs of sequential approximate solutions of Warga) as filters 
of 
with the property 
Moreover, we can be restricted to the employment of only ultrafilters (maximal filters) with the above-mentioned 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 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 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).
2. 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 
and
, 
is the unordered pair of x and y (see [10]). Then, 
is singleton containing an object x. Of course, for any objects x and 
the object 
is the ordered pair of objects x and y; here, we follow to [10]. By 
we denote the empty set. By a family we call a set all elements of which are sets.
By 
we denote the family of all subsets of a set 
then, 
are the family of all nonempty subsets of X. Of course, for any set 
in the form of 
and 
we have the family of all nonempty subfamilies of 
and 
respectively.
If 
is a set, then we denote by 
the family of all finite sets of 
then 
is the family of all finite subsets of 
For any sets 
and 
we denote by 
the set of all mappings from 
into 
If 
and 
are sets, 
and 
then
(the image of 
under the operation f) and 
is the usual 
-restriction of 
. In the following, 
and 
is the real line; 
Of course, we use the natural order 
of
. If
, then
Transformations of families. For any nonempty family 
and a set 
we suppose that
If X and Y are sets and
, then we suppose that
(2.1)
of course, in (2.1) nonempty families are defined.
If 
is a family, then we suppose that
(we keep in mind that 
is a nonempty set 
and, for
, 
is a family) and
(of course, for
, 
is a nonempty family); moreover
So, for any nonempty family
, we obtain that
of course,
.
Special families. Let I be a set. Then, we suppose that
(2.2)
elements of (2.2) are called 
-systems with “zero” and “unit”. Moreover,
(2.3)
elements of (2.3) are lattices of subsets of I (with “zero”). Finally,
(2.4)
Of course, in (2.4) lattices of sets with “zero” and “unit” are introduced. We note that
Of course,
(2.5)
is the set of all topologies of I. If 
then the pair 
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 
is the mapping for which
(2.8)
From (2.5) – (2.8), we obtain the following properties:
(2.9)
We note that 
in addition,
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 
then 
is a measurable space with an algebra of sets.
If 
and 
then by 
we denote the set of all mappings
for each of which:
1) 
2) 
Then
(2.11)
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 
has the properties: 1) 
2) 
Now, we introduce some notions important for constructions of general topology. Namely, we consider topological bases of two types:
(2.12)
(2.13)
Of course,
In connection with (2.12), we suppose that
,
;
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:
(2.16)
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)):
(2.17)
Of course, in (2.17), we use the usual duality property connected with (2.14) – (2.16).
Some additions. In the following, we suppose that
(2.18)
if 
then TS 
is called 
-space. We use (2.18) under investigation of properties of topologies on ultrafilter spaces.
Finally, we suppose that
. So, we introduce “continuous” lattices.
3. Nets and Filters as Approximate Solutions under Constraints of Asymptotic Character
In this section, we fix a nonempty set 
considered (in particular) as the space of usual solutions. We consider families 
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
: in the set 
the set of 
-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 
(by 
the set of all families 
(families
) for which
Then, 
is the set of all filter bases on
. By 
we denote the set of all filters on
:
(3.1)
Using (3.1), we introduce the set 
of all ultrafilters on
:
(3.2)
In connection with (3.1) and (3.2), see in particular [11, ch. I]. In addition,
(3.3)
By (3.3) we define the filter on 
generated by a base of 
If 
then by 
(by 
we denote the set of
all filters (ultrafilters 
) such that
. Then, for any filter 
, we have 
and what is more 
is the intersection of all ultrafilters 
see [11].
If a family 
is considered as the constra int of asymptotic character, then ultrafilters 
are considered as (nonsequential) approximate solutions; of course, filters 
can be considered in this capacity also. But, ultrafilters have better properties; therefore, now we are restricted to employment of ultrafilters as approximate solutions.
The filter of neighborhoods. If 
and
, then
and 
of course, 
in correspondence with (3.3). We were introduce the filter of neighborhoods of 
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]. Suppose that 
(3.5)
In addition, 
see (3.1). Therefore, we can use (3.5) in the case of 
where 
we note that 
Then, by (3.5) 
(3.6)
Of course, it is possible to use the variant of (3.6) corresponding to the case 
where 
Nets and the Moore-Smith convergence. On the basis of (3.6), we can to introduce the standard MooreSmith convergence of nets. We call a net in the set 
arbitrary triplet 
where 
is a nonempty DS and 
If 
is a net in the set 
then
(3.7)
we obtain the filter of 
associated with 
Now, for any topology 
a net 
in the set 
and 
we suppose that
(3.8)
From (3.6) and (3.7), we obtain that (3.8) is the “usual” Moore-Smith convergence (see [12]). Of course, any sequence 
generates the net 
where 
is the usual order of
.
If
, then a net 
in 
is called 
-admissible if 
In this case, 
can be considered as a constraint of asymptotic character and 
plays the role of nonsequential (generally speaking) approximate solution.
In conclusion, we note that
(3.9)
In (3.9), trivial ultrafilters are defined.
4. 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 brevity, in this section, we fix following two nonempty sets: 
and 
In addition, under 
and 
(4.1)
of course, in (4.1), we can use a filter or ultrafilter instead of
. In addition, the important property takes place: if 
and
, then
(4.2)
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.
Introduce AS: if 
and
, then by 
we denote the set of all 
for each of which there exists a net 
in the set X such that
(4.3)
we consider 
as AS. In this definition, we use nets. But, for any filter 
there exists a net 
in the set X for which 
(see [13]).
Proposition 4.1. For any 
and 
(4.4)
Proof. Fix 
and 
Suppose that 
and 
are the sets on the left and right sides of (4.4) respectively. Let
. Then 
and, for a net 
in X, the relation (4.3) is valid under 
Then, by (4.3)
(4.5)
Moreover, by (3.8) and (4.3) 
So, by (3.6)
(4.6)
Let 
Then by (3.7) and (4.6), for some 
the following property is valid: 
(4.7)
In addition, 
and by (3.7) and (4.5)
As a corollary, 
But, by (4.7) 
By (3.3) 
Since the choice of 
was arbitrary, the inclusion 
is established. By (3.5)
(4.8)
By (4.5) and (4.8) 
The inclusion 
is established.
Let 
Then, for 
we have a filter 
such that
(4.9)
Choose a net 
in X for which
. By (4.1) 
and, as a corollary, by (3.5) and (4.9)
(4.10)
Then by (3.3) and (4.10) we obtain that
. Using (2.1) we have the property: 
Choose arbitrary 
then, for some 
the inclusion 
is valid. By (3.7) and the choice of 
for some 
the following property is realized: 
By the choice of 
we obtain that 
Then, 
So, the important inclusion
is valid. Then 
(see (3.6)). By (3.8)
(4.11)
Moreover, by the choice of 
and 
the inclusion
is valid. From (4.11), we have the inclusion 
So, 
and, as a corollary, 
Proposition 4.2. For any 
and 
(4.12)
Proof. We denote respectively by 
an
the sets on the left and right sides of (4.12). Since
, we have the obvious inclusion 
(see Proposition 4.1). Let 
Then by Proposition 4.1 
for some 
Then 
and 
We recall (see Section 3) that
. Choose arbitrary 
Then 
and 
Therefore, 
Moreover, by (2.1) 
and, as a corollary,
(4.13)
(we recall that by (4.1) 
and 
). By the choice of 
we have the inclusion
(see (3.5)). Then by (4.13) 
and, as a corollary (see (3.5)),
Then, 
The inclusion 
is established.
Recall that, for any family 
and 
We note the following obvious.
Proposition 4.3. For any
, the equality 
is valid.
Proof. Recall that 
Therefore,
; on the other hand, from (3.1), we obtain that 
Then, for an ultrafilter 
and, as a corollary, 
So, since the choice of 
was arbitrary, 
and, as a corollary, 
Corollary 4.1. If
and
, 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. Namely, 
(4.15)
Then, by (4.14), (4.15), and Corollary 4.1
(4.16)
In connection with (4.16), we note that 
(4.17)
Remark 4.1. By analogy with Proposition 4.3 we have that
Really, fix 
Then 
Therefore, 
Let
. Then, 
and 
But, from (3.1), we have the equality 
where by the choice of
. So, 
and, as a corollary, 
The inclusion 
is established. So, 
Returning to (4.17), we note that by Proposition 4.2 
(4.18)
Remark 4.2. We have that, for the case 
it is possible that
Indeed, consider the case 
is the usual 
-topology of real line
, and
Then, 
and 
But, by (4.15)
It is obvious the following.
Proposition 4.4. If 
and 
then
Proof. The corresponding proof follows from known statements of general topology (see [11]). But, we consider this proof for a completeness of the account. In our case, we have (4.15). In addition,
(4.19)
is nonempty family of sets closed in the compact topological space (TS) 
Moreover, 
(we use known properties of the closure operation and the image operation). Since 
we obtain that 
In addition, 
Therefore, by [9] we have the following property: if 
and
then 
As a corollary, 
is the non empty centered system of closed sets in a compact TS. Then, the intersection of all sets of 
is not empty. By (4.19)
Using (4.15), we obtain the required statement about the nonemptyness of attraction set.
Corollary 4.2. If 
and 
then
Proof. Let 
Choose arbitrary topology 
By (4.14) 
Moreover, 
Therefore, 
Then, 
and by Proposition 4.4
Using Corollary 4.1, we obtain that 
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
(4.21)
In (4.21), the set of all bijections from 
onto 
is defined. If 
and 
then
(4.22)
(4.23)
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.
5. Some Properties of Ultrafilters of Measurable Spaces
In this section, we fix a nonempty set
. We consider the very general measurable space 
where 
is fixed also. According to necessity, we will be supplement the corresponding suppositions with respect to
. We suppose that 
is the set of all families 
such that
Elements of the set 
are filters of
. In addition,
(5.1)
is the set of all ultrafilters of 
Recall that (see [16])
(5.2)
In the following, (5.2) plays the very important role.
We introduce the mapping 
by the following rule:
(5.3)
We note that 
and by (5.2) 
In addition, we recall that (see Section 2)
(5.4)
by (5.4) the pair (
) is a nonempty multiplicative space. We note some simplest general properties. We obtain that
We note that
In addition, for 
the inclusion 
takes place. Therefore,
(5.5)
With the employment of (5.5), we obtain that, for any 
and 
(5.6)
Now, we return to the space 
Suppose that
(5.7)
(the set of filter bases of
); 
and
We note the obvious property: 
In addition,
Using (5.5) and the obvious inclusion 
under 
and 
we obtain, that 
We note that, under 
and 
the filter
has the following properties
(5.8)
Of course, 
We can use this property in (5.8): for any 
and 
the filter 
has the properties
(5.9)
In connection with (5.9), we recall the very general property: if 
and 
then 
Using the maximality property, we obtain that
And what is more, 
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 
and 
then (under condition (5.10))
(5.11)
The property (5.11) is basic. As a corollary, 
(5.12)
We note that by (5.11) the following property is valid:
As a corollary, we obtain the property
(5.13)
(so, under (5.10), the statement (5.4) is amplified). In (5.13), we have the lattice of subsets of 
This important fact used below.
6. Topological Properties, 1
As in the previous section, now we fix a nonempty set 
and a family 
We note the following obvious property:
From definitions of the previous section, the following known property follows: 
(6.1)
Moreover, we note that 
(6.2)
Moreover, we note that
. Therefore, by (5.4)
(6.3)
As a corollary, we obtain (see Section 2) that
(6.4)
We recall the very known definition of Hausdorff topology; namely, we introduce the set of such topologies: if 
is a set, then
For any set M we suppose that
If 
then TS 
is called a compactum. Then, the obvious statement follows from the ultrafilter properties (see (5.3), (6.1)):
(6.5)
So, by (6.5) 
is a Hausdorff TS. Of course, we can use the previous statements of this section in the case of 
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 
and the sets 
and 
are defined. From (3.1) and definitions of Section 5, we have the equality 
Moreover, from (3.2) and the above-mentioned definitions of Section 5, the equality
(6.6)
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 chain 
So, by (3.3)
In particular, we have the following property:
(6.7)
We note one general simple property; namely, in general case of 
(6.8)
Remark 6.1. We note that (6.8) is a variant of Proposition 2.4.1 of monograph [16]. Consider the corresponding proof. Fix 
Then by (6.7)
(6.9)
From (6.9), we obtain (see Section 3) that 
Let
Then, 
and 
In addition (see Section 5), 
Let 
Then, 
and, in particular, 
By (6.9) 
and, as a corollary, 
Then, 
So, the inclusion 
is established; we obtain that
(6.10)
From (5.1) and (6.10), we have the equality 
So,
Since the choice of 
was arbitrary, the property (6.8) is established.
7. Topological Properties, 2
In this and following sections, we fix a nonempty set 
and a lattice 
We consider the question about constructing a compact 
-space with “unit” 
This space is similar to Wallman extension for a 
-space. But, we not use axioms of topology and operate lattice constructions (here, a natural analogy with constructions of [14] takes place). Later we use the following simple statement.
Proposition 7.1. 
Proof. We use (5.13). In particular, 
As a corollary,
(7.1)
Moreover, 
(see (5.4)). So, 
is a family with “zero” and “unit”. Moreover, by (5.13)
Therefore, by (2.13) the required statement is realized.
By (2.15) and Proposition 7.1 we have the following construction:
(7.2)
Proposition 7.2. The following compactness property is valid:
(7.3)
Proof. For brevity, we suppose that
(7.4)
and 
Of course, by (2.9) 
Moreover, under 
the family
has the following obvious property
(7.5)
We have the equality 
So, 
is the family of all subsets of 
closed in the TS
(7.6)
Let 
be arbitrary nonempty centered subfamily of 
(for any 
and 
the intersection of all sets 
is not empty). If 
then the family
(7.7)
has the property: 
Of course,
is centered. Indeed, choose 
and 
Let 
be a procession with the property:
Then, in particular, 
In addition, by (7.7)
Of course,
Since the intersection of all sets 
is not empty (we use the centrality of
), we choose an ultrafilter
Then, 
under 
By axioms of a filter (see Section 5) we obtain that
Since 
is closed with respect to finite intersections, we obtain that
(7.8)
Moreover, (7.8) is supplemented by the following obvious property; namely,
From (5.7), we obtain that 
As a corollary,
in addition, by (7.8) 
Finally, we use (5.2). Let 
be an ultrafilter for which 
Then, 
So,
(7.9)
Let 
Then, 
and the equality
(7.10)
is valid (see (7.5)). Choose arbitrary 
Then, 
and 
Using (5.4), we choose 
for which 
Then
By (7.7) 
and, in particular,
. By (7.9) 
and, as a corollary, 
see (5.3). So, 
Since the choice of 
was arbitrary, we obtain that 
By (7.10) 
So, we have the property:
Then, the intersection of all sets of 
is not empty. Since the choice of 
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 [11-13]).
Using Proposition 7.2, by 
we denote the topology (7.3); so,
(7.11)
We have the nonempty compact TS
(7.12)
Proposition 7.3. If 
then
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:
(7.13)
So, by (7.13) we obtain that (7.12) is a nonempty compact 
-space.
In conclusion of the given section, we note several properties. First, we recall that
(7.14)
In addition, from (7.11), the obvious representation follows:
(7.15)
With the employment of (7.15) the following statement is established.
Proposition 7.4. If 
then the family
is a local base of TS (7.12) at
:
The proof is obvious. So, by (3.4) and Proposition 7.4
We note that, from definitions, the following property is valid:
(7.16)
8. 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
. But, in this section, we postulate that 
So, in this section
(8.1)
unless otherwise stipulated. So, 
and
. Therefore, with regard (3.9) and (8.1), we obtain that
(8.2)
Of course, for any 
the inclusion 
is valid.
Proposition 8. 1.
Proof. Let 
and 
We use Propposition 7.4. Namely, we choose a set 
for which
(8.3)
Since 
by axioms of a filter (see Section 5), we obtain that 
in addition, 
and by (2.4) and (8.1) 
So, 
Choose arbitrary point 
and consider the ultrafilter
(8.4)
see (8.2). In addition, 
As a corollary, by definitions of Section 5
(8.5)
But, 
by the choice of e. Therefore, by (8. 5) 
From (5.3) we have the property 
As a corollary, by (8.4)
(8.6)
From (8.3) and (8.6), we obtain that 
By (8.4)
(8.7)
Since the choice of 
was arbitrary,
Since the choice of 
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, 
then, for 
And what is more by (2.17), (7.11), and Proposition 7.1, in general case of 
(8.8)
so, by (8.8) 
is a base of topology (7.11). We recall that by (2.8) and (5.4), for general case of 
(8.9)
Connection with Wallman extension. Let 
Then, 
and by (2.18) 
Using (2.7), we obtain that 
with the employment of the above-mentioned closedness of singletons, by the corresponding definition of Section 2 we obtain that
(8.10)
Until the end of the present section, we suppose that
(8.11)
So, in our case, 
is the lattice of closed sets in T1- space. Then, (7.12) is the corresponding Wallman compact space (see [13]). On the other hand, by (8.10) and (8.11) we obtain that this variant of 
corresponds to general statements of our section (for example, see (8.2) and Proposition 8.1). In this connection, we consider the mapping
(8.12)
we denote the mapping (8.12) by
. So, 
and
Consider some simple properties. First, we note that 
is injective: 
(8.13)
Indeed, for 
and 
with the property 
by (3.9) we have that 
and, as a corollary, 
so, 
Of course, 
is a bijection from E onto the set
(8.14)
If 
and 
then 
As a corollary, we obtain that
(8.15)
Remark 8.1. Of course, in (8.15), we use the representation (8.12). Fix 
Let 
Then 
and 
By (5.3) 
and, as a corollary, 
So,
(8.16)
If 
then 
see (8.12). Therefore, by (5.3) 
and, as a corollary, 
So, 
Therefore (see (8.16)) 
and 
coincide.
From (5.4) and (8.15), we obtain that
(8.17)
Proposition 8.2. 
Proof. We use the construction dual with respect to (4.20). Let 
Then, by (8.8) 
Therefore, for some 
As a result, we obtain that
(8.18)
where 
see (8.17). By (2.6), (2.9), (8.11), and (8.18) we have the property:
and
(8.19)
Since the choice of F was arbitrary, from (8.19) we obtain the required continuity property (see [16, (2.5.2)]).
Corollary 8.1. 
Proof. Recall that 
In addition, by (8.14)
Let 
and 
realizes the equality 
By Proposition 8.2
(8.20)
In addition, 
(indeed,
). Let 
Then, 
and 
But, 
too. Then, 
So, 
Therefore, 
Since the choice of 
was arbitrary, the inclusion
is established. So, 
By (8.20) 
Since the choice of G was arbitrary, the inclusion
is established.
Recall that 
(see (4.21)).
Proposition 8.3. 
Proof. Let 
Then 
and 
By (8.11) 
In addition, by (5.3)
(8.21)
Of course, by (5.4) 
Then
As a corollary, 
Therefore,
(8.22)
Now, we compare 
and 
(8.22). Let 
Then, for some 
(8.23)
Of course, 
By (3.9) 
(indeed,
). By (8.23) 
and, as a corollary, 
see (8.21). We obtain that
(8.24)
Since 
we have the inclusion 
Using (8.22) and (8.24), we obtain that 
The inclusion
(8.25)
is established. Choose arbitrary 
then, by (8. 22), for some 
the equality 
is valid. So,
(8.26)
Moreover, 
So, 
By (8.21) 
Since 
by (8.26) 
From (3.9), the property 
follows. Then, 
Therefore, 
as a corollary, 
The inclusion 
is established. Using (8.25), we obtain that 
By (8.22)
Since the choice of G was arbitrary, by Corollary 8.1 and (4.22) we have the inclusion
By (4.24), (8.13), and Proposition 8.3 we obtain that
(8.27)
So, we construct the concrete homeomorphic inclusion of 
-space in the compact 
-space (in this connection, we recall that by Proposition 8.1
moreover, see (7.13)). So, we have the “usual” Wallman extension.
9. Ultrafilters of Measurable Space
In this section, we fix a nonempty set 
and an algebra 
of subsets of 
So, in this section, 
is a measurable space with an algebra of sets: 
Of course, we can to use constructions of Section 5; indeed, in particular, we have the inclusion 
see (2.10). As a corollary, by (2.4) 
So, we use the sets 
and 
of Section 5; we use properties of these sets also. We note the known representation (see [15]):
(9.1)
Now, we use (9.1) for investigation of TS (7.12) in the case 
First, we note the obvious corollary of (9.1):
(9.2)
Remark 9.1. Let 
is fixed. Choose arbitrary 
Then, by (7.14) 
By (9.1) 
where 
by axioms of an algebra of sets. So, by (5.3) 
The inclusion
(9.3)
is established. Let 
Then, by (5.3) 
and 
By axioms of a filter
So, 
and 
As a corollary, 
So, the inclusion
is established. Using (9.3), we obtain the required coincidence 
and 
Returning to (9.2) in general case, we note the following obvious Proposition 9.1. 
Proof. Let 
Using (5.4), we choose 
such that 
Then 
and by (9.2)
(9.4)
From (5.4), we have the obvious inclusion 
By (9.4)
Therefore, we obtain the following property:
The inclusion 
is established. Choose arbitrary
(9.5)
Using (2.8), we choose 
such that 
Let 
be the set for which 
see (5.4). Then, by (9.2)
(9.6)
where 
Since by (5.4) 
from (9.6), we obtain that
Since the choice of 
(9.5) was arbitrary, the inclusion
is established. So, we obtain the required equality.
From (6.4), (8.8), and Proposition 9.1, the simple (but useful) statement follows.
Proposition 9.2. 
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
(9.7)
So, we obtain a nonempty compactum. Recall that (see (7.11), Proposition 9.2)
(9.8)
is the family of all sets closed in the sense of topology (9.7). We note the following obvious property (see [15, ch.I])
(9.9)
Remark 9.2. We recall (5.4). Let 
Using (5.4), we choose 
such that 
Then, 
and by (9.2)
(9.10)
By (5.4) and (9.10) 
So, we establish that
(9.11)
From (2.10), (5.4), and (9.11), the property (9.9) follows.
Proposition 9.3. 
Proof. Recall that by statements of Section 2 and (9.8) the inclusion
(9.12)
From (6.4), the inclusion 
follows too. So, by (9.12)
(9.13)
Let 
Since 
is open, then by (6.4) we obtain that, for some family
(9.14)
the following equality is realized:
(9.15)
If 
then by (9.15) 
and, as a corollary, 
where 
So, by (5.4) we obtain the implication
(9.16)
Let 
Then, 
Since 
is a closed subset of a compactum, we have the compactess property of
; then, by (9.14), for some 
(9.17)
In particular, 
We note that 
is closed with respect to finite unions (indeed, by (9.9) 
is an algebra of sets). Therefore, by (9.17) 
in the case 
So,
(9.18)
Using (9.16) and (9.18), we obtain that 
in any possible cases. Since the choice of 
was arbitrary, the inclusion
(9.19)
in established. From (9.13) and (9.18), the required statement follows.
So, 
is the family of all open-closed sets in the nonempty compactum
(9.20)
In connection with the above-mentioned property of nonempty compactum (9.20), we recall [15, ch. I]. With the employment of (9.1), the following obvious property is established: in our case of measurable space with an algebra of sets
(9.21)
Remark 9.3. For a completeness, we consider the scheme of the proof of (9.21). For this, we note that by (3.9) and the corresponding definition of Section 5
(9.22)
In particular, by (9.22) 
Fix 
and suppose that
of course, 
In addition, 
Then, 
(9.23)
Of course, by (3.9), for
, we have the following obvious implications:
Then, by (9.23) 
Since the choice of 
was arbitrary, by (9.1) 
So, (9.21) is established.
Using (9.21), we introduce the mapping
(9.24)
Of course, in (9.24) we have analog of the mapping 
(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 
is distinguishing for points of
.
If 
then by analogy with Section 4 we suppose that
(9.26)
of course, 
and moreover the following property is valid:
(9.27)
Returning to (9.25), we note that
(9.28)
In (9.28), we can use 
as constraints of asymptotic character. Of course, 
(see Section 5). Then, by (3.3)
(9.29)
By analogy with (9.29) we note that 
and 
These properties permit realize an asymptotic analogs of solutions of the set (9.28). In this capacity, we can use elements of the sets 
and 
where 
is used as “asymptotic constraints”. Of course, 
bounds our possibilities: we can use only subfamilies of
.
Proposition 9.4. 
Proof. Fix 
Let 
Then 
So, 
and 
Choose arbitrary 
Then, by (9.24)
(9.30)
By the choice of 
we have the inclusion 
Since 
we obtain that 
Then, by (9.30) 
Since 
by (5.3)
(9.31)
By (9.30) and (9.31) we obtain the following property
Since the choice of 
was arbitrary, we have (see (8.3)) the statement
(9.32)
Choose arbitrary 
Then, for some 
the inclusion 
is valid. Therefore, 
and 
By (6.4), there exists 
such that
(9.33)
From (9.32), the property 
is valid. By (9.33) we obtain that
(indeed,
). Since the choice of 
was arbitrary,
Then, 
So, the inclusion
is established. The opposite inclusion is obvious.
We note that Proposition 9.4 is similar to Proposition 8.1. But, in the given section, the condition
(9.34)
is supposed not; in Section 8 (in particular, in Proposition 8.1), the condition similar to (9.34) is essential. So, Proposition 9.4 has the independent meaning.
10. Attraction Sets Under the Restriction in the Form of Algebra of Sets
In the following, we fix a nonempty set E, a TS 
where 
and a mapping 
Elements 
are considered as usual solutions and elements 
play the role of some estimates. The natural variant of an obtaining of 
is realized in the form 
where 
But, we admit the possibility of the limit realization of 
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 
Then, we obtain constructions of Section 4 under 
and 
But, we admit yet one possibility: along with “usual” AS, we use the sets
(10.1)
Of course, we use remarks of the conclusion of the previous section.
Proposition 10.1. If 
and 
then
(10.2)
Proof. We use reasoning analogous to the proof of Proposition 4.2. We denote by 
the set on the right side of (10.2). Since 
(see Section 9), by (10.1)
(10.3)
Let 
Then, by (10.1) 
and, for some 
(10.4)
Recall that 
(see Section 9). Therefore, by (4.1) 
Then, (10.4) denotes that
(10.5)
(see (3.5)). In addition, by the choice of 
we have the inclusion 
see (9.26). By (9.27), for some 
the inclusion 
is valid. Then,
As a corollary, by(3.3) and (10.5)
where 
(see Section 9). Then, by (3.5)
(10.6)
By definition of 
we obtain that 
Since the choice of 
was arbitrary, the inclusion
(10.7)
is established. Using (10.3) and (10.7), we obtain the required equality
(10.8)
From the definition of 
and (10.8), we obtain (10.2).
Recall that 
and therefore
By definitions of Section 3, (6.6), and (9.26) we obtain that
(10.9)
From Propositions 4.2 and 10.1, we have (see (10.9)) the property:
So, our new construction is coordinated with AS of Section 4. Moreover, under 
we can consider AS 
for 
Proposition 10.2. If 
and 
then
(10.10)
Proof. We use (6.8). Choose 
Then, 
and, for some 
the convergence
(10.11)
is valid. Then, 
and 
see (9.26). By (6.8) for some 
the equality 
is valid. Then, 
As a corollary, 
Now, we return to (10.11). In addition, 
Therefore, 
and by (3.3)
From (3.5) and (10.11), we have the obvious inclusion
(10.12)
In addition, 
and 
see (4. 1). Since 
the inclusion 
is valid. As a corollary, by (3.3)
Using (10.12), we obtain the basic inclusion
(10.13)
From (3.5) and (10.13), we obtain the following convergence
(10.14)
So, 
has the property (10.14). Then, by Proposition 4.2
Since the choice of 
was arbitrary, the required inclusion (10.10) is established.
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 
So, 
is a measurable space with an algebra of sets. In this case, we can supplement the property (6.8). Namely,
(10.15)
Remark 10.1. We omit the sufficiently simple proof (10.15). Now, we are restricted to brief remarks. Namely, by ultrafilter 
we can realize a finitely additive (0,1)-measure 
on the family 
supposing that 
under 
and 
under 
In connection with such possibility, we use [9,(7.6.17)] (moreover, see [9,(7.6.7)]). The natural narrowing v of 
on our algebra 
is finitely additive (0,1)-measure on 
(of course,
). Therefore, for some 
by [9,(7.6.17)] 
is defined by the rule
(10.16)
On the other hand, the family 
realizes 
by the obvious rule:
(10.17)
From (10.16) and (10.17), the required equality 
follows. Then, by the choice of 
we have the inclusion 
Using (6.8) and (10.15), we obtain that
(10.18)
By (10.18) we establish the natural connection of 
and 
Now, we consider some other auxiliary properties.
If 
and 
then we have the following equivalence
(10.19)
Of course, we can use instead of 
the corresponding image of a filter base in 
Indeed, by (4.1) and (10.19) 
(10.20)
Moreover, in connection with (10.20), we note that 
(10.21)
Remark 10.2. Consider the proof of (10.21). Fix 
and 
Let 
Then, by (10.20)
Therefore, for any
, there exists 
such that 
As a corollary,
Then, 
Since the choice of 
was arbitrary,
So, 
Let
(10.22)
Choose arbitrary neighborhood 
Then, by (10.22) 
Therefore, for some 
the inclusion 
is valid. In addition, 
and
Then, 
Therefore, 
and by (10.20) 
So,
The proof of (10.21) is completed.
We note that, in (10.21), we can use instead of 
arbitrary filter of 
In this connection, we recall that by constructions of Section 5, for any 
we obtain (in particular) that 
and
(10.23)
Then, from (10.21) and (10.23), we have the following property: 
(10.24)
Of course, (10.24) is the particular case of (10.21); in (10.23), we have the useful addition. We note that 
(10.25)
Remark 10.3. Fix 
and 
Consider the proof of (10.25). By (3.4) and (3.5) we have the following implication
(10.26)
Let 
Choose arbitrary 
Then, by (2.18), for some 
the inclusion 
is valid. Since 
by filter axioms (see 3.1)) 
So, the inclusion 
is established. By (3.5) we have the convergence 
So,
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)): 
(10.27)
Remark 10.4. Consider the proof of (10.27). We fix 
and 
Since 
(see (3.4) and definitions of Section 3), by (10.21)
(10.28)
Let the corollary of (10.28) is valid. Fix 
with the property
(10.29)
Let 
Then, by (3.4), for some 
the inclusion 
is valid, where 
By (10.29) 
and 
From (3.1) and (3.3), the inclusion 
follows. Since the choice of 
was arbitrary, the inclusion
is established. By (10.21) 
So, we obtain that
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): 
(10.30)
Remark 10.5. Consider the proof of (10.30), fixing 
and 
Then, by (10.15) 
In particular (see Section 9), 
Now, (10.30) follows from (10.27).
Condition 10.1. 
Remark 10.6. It is possible to consider Condition 10.1 as a weakened variant of the measurability of 
The usual measurability of 
is not natural since 
is only algebra of sets.
Until the end of the present section, we suppose that Condition 10.1 is valid.
Proposition 10.3. If Condition 10.1 is fulfilled, then 
Proof. Let Condition 10.1 be fulfilled. Fix 
and 
Then, 
and, for some 
(10.31)
(see Proposition 4.2). Then, by (10.21) and (10.31) we have the inclusion 
since 
by (3.1). As a corollary,
(indeed, for 
we can choose 
such that 
therefore, by (2.1) 
and by (3.1)
). By Condition 10.1 there exists 
such that 
In addition,
(10.32)
therefore, 
where by (10.15) 
We recall that 
(see Section 9) and
(10.33)
Of course, by (4.1) 
In addition, by (10.33)
By (10.27) 
Recall that 
Since 
we obtain that 
Therefore (see (9.26)),
By Proposition 10.1 
Since the choice of 
was arbitrary, we have the inclusion
(10.34)
Using (10.34) and Proposition 10.2, we obtain the equality
So, we can use (see Proposition 10.1 and Condition 10.1) ultrafilters of the space 
as nonsequential approximate solutions in the case, when a nonempty subfamily of 
is used as the constraint of asymptotic character. This property is very useful in the cases of spaces 
for which the set 
is realized effectively. In addition, for a semialgebra 
with the property 
(see Section 2), we consider the passage
as an unessential transformation (see [9,§7.6] and [16,§ 2.4]; here it is appropriate to use the natural connection of ultrafilters and finitely additive (0,1)-measures). Then, after unessential transformations, the examples of [9,§ 7.6] can be used in our scheme sufficiently constructively.
11. Ultrasolutions
First, we recall some statements of [17]. In addition, we fix a nonempty set E and a TS 
where 
We consider the nonempty set 
Suppose that 
Then, we suppose that
(11.1)
So, we introduce the limit sets corresponding to ultrafilters of 
By analogy with Proposition 5.4 of [17] the following statement is established.
Proposition 11.1. If 
then 
Proof. Fix 
Then 
and by (4.2)
(11.2)
(recall that 
Since 
is a compact TS, there exists 
such that 
see [9, ch. I]. Then, by (3.6) 
or
(11.3)
(see (11.2)). By (3.5) and (11.3) 
Then, by (11.1) 
So,
(11.4)
Let 
Then 
and 
By (3.5) and (11.2) 
In addition, by (11.2) 
Then, by (3.1)
By (2.1) we obtain that
Since 
and 
for 
we have the property:
So, 
The inclusion
is established. Using (11.4), we obtain that 
We note the following obvious property too: if 
and 
then
(11.5)
Remark 11.1. Let the premise of (11.5) be fulfilled. Then, by (3.6)
(11.6)
Then, 
Indeed, suppose the contrary: 
Then, by (6.1), for some 
and 
the equality 
is valid. But, by (11.6) 
and 
Then, by (3.1) 
The obtained contradiction means that 
is impossible. So, 
Proposition 11.2. If 
and 
then
Proof. The corresponding proof is the obvious combination of (11.1), (11.5), and Proposition 11.1. Indeed, by Proposition 11.1 
and 
Let 
Then, 
and
(11.7)
Let 
Then, 
and
(11.8)
For
, by (11.7) and (11.8)
So, by (3.6) 
and 
From (11.5) the equality 
is valid. Then, 
The inclusion
(11.9)
is established. But, by the choice of 
we have the inclusion 
Using (11.9), we obtain that
The uniqueness of 
is obvious.
From Proposition 11.2 the natural corollary follows: if 
then
(11.10)
In the following, we postulate that
(11.11)
Then (see (11.10) and (11.11)), we suppose that
(11.12)
is defined by the following rule: if 
then 
has the property:
(11.13)
We note that by (11.13) and Proposition 11.1
(11.14)
So, by (11.12) and (11.14) 
In this connection, we note the following typical situation: under condition (11.11), 
and 
Of course, by (11.11)
Indeed, any closed set in a compact TS is compact too. Recall that
(11.15)
Returning to (11.1) and (11.13) we note that
(11.16)
With the employment of (3.9), we introduce the natural immersion of E in 
supposing that
(11.17)
In connection with (11.17), we note the following obvious equality:
(11.18)
Remark 11.2. Consider the proof of (11.18). Fix 
By (3.9) we obtain that
(11.19)
So, by (2.1) and (11.19) we obtain the following property:
(11.20)
In addition, by (3.5), (3.9), and (11.16)
So,
. Then, by (11.20) 
Using the separability of 
(11.11), we obtain the equality chain
Since the choice of 
was arbitrary, we obtain that (11.18) is fulfilled.
Proposition 11.3. If 
then the following equality is valid:
Proof. Let 
Then 
and by (11.16)
Then, by (3.5) 
Therefore, for any 
there exists 
such that 
(see (3.3) and (4.1)).
Let 
Then, 
If 
then, for some 
the inclusion 
is valid; moreover, 
and
(11.21)
where 
So, 
Since the choice of 
was arbitrary, we obtain that
Therefore, 
Since the choice of 
was arbitrary too, we have the inclusion 
Therefore,
Choose arbitrary 
Then 
and
(11.22)
Then, for 
we obtain the property 
Using (3.3), we have the following statement:
Therefore, 
(see Section 5), where
(11.23)
(we use (4.2)). In addition, using (6.6), (11.23), and statements of Section 5, we obtain that
Therefore, 
Since the choice of 
was arbitrary, we obtain that
So, by (3.5) 
Then, we have the following properties:
By (11.5) 
Then 
Since the choice of 
was arbitrary, we obtain that
The opposite inclusion was established previously. Therefore, 
and the intersection of all sets 
coincide.
From (4.15) and Proposition 11.3 we obtain that
(11.24)
So, ultrafilters of 
realize very perfect constraints of asymptotic character.
12. Ultrafilters of Measurable Space with Algebra of Sets
In this section, we fix a nonempty set E, TS 
and 
Moreover, we fix 
Finally, we suppose that Condition 10.1 is fullfiled. Then, we have the statement of Proposition 3 and other statements of Section 10. We suppose that (11.11) is valid also. So, we have the mapping (11.12). In addition, we have the natural uniqueness of the filter limit: (11.5) is fulfilled. Now, we supplement (11.5). Namely, 
(12.1)
Remark 12.1. For the proof of (12.1), we fix 
and 
Let the premise statement of (12.1) is valid: 
converges to 
and
. Then, for 
(see (3.3)), the inclusions
are fulfilled. Therefore, by (3.6) the following two properties are valid:
By (11.5) 
So, (12.1) is established.
We recall (10.23) and (10.24): if 
then 
and 
(see (4.1)). We use (10. 15).
Proposition 12.1. 
Proof. Let 
and 
Then 
and by Condition 10.1, for some
, the inclusion 
is fulfilled. In addition, by (11. 16) 
Since 
and 
then
(12.2)
From (12.2) the inclusion 
follows (namely, for any 
there exists 
such that 
then, 
by (12.2) and 
by axioms of a filter). In addition, 
Then,
and (by the choice of 
Since 
by (10.27)
By definition of 
Proposition 12.2. 
Proof. Fix 
Using (10.18), we choose 
such that
(12.3)
Then, by (11.12) 
and by (12.3) and Proposition 12.1
(12.4)
In addition, 
and 
Therefore, by (12.1) and (12.4) 
From Proposition 12.2, the obvious corollary follows; namely 
Now, we suppose that the mapping
(12.5)
is defined by the following rule: if 
then
(12.6)
From (10.15) and (12.5), the obvious property follows; namely, 
Proposition 12. 3. 
Proof. Fix 
Then, by (11.12) 
By (10.15) we obtain that 
In particular, 
and by (4.1) 
From Proposition 12.1, we have the following convergence
(12.7)
Using Proposition 12.2, (12.1), (12.6), and (12.7), we obtain that 
Proposition 12.4. If 
and 
then 
Proof. Using (10.18), we choose 
such that 
Then, by Proposition 11.3 we have the inclusion
(12.8)
(we use the obvious inclusion 
realized by the choice of
). By Proposition 12.3
From (12.8), the inclusion 
follows.
We note that (see [11,12,13]) by (11.11) the space 
is regular: if 
then
(12.9)
Proposition 12.5. The mapping (12.5) is continuous:
(12.10)
Proof. Fix 
Then, by (12.5) 
In addition, by (12.6)
(12.11)
Of course, 
and 
(see (4.1)). As a corollary, by (3.3)
(12.12)
From (3.5), (12.11), and (12.12), we obtain the following inclusion:
(12.13)
From (2.1), (3.3), and (12.13), we obtain that
(12.14)
Fix 
Using (12.9), we choose 
such that 
Then, by (3.4), for some 
the inclusion
(12.15)
is valid. Therefore, 
Of course, 
Therefore, by (12.14), for some 
(12.16)
In addition, 
(see (5.4)). By (6.4) 
In addition, by (5.3) 
Therefore,
(12.17)
Choose arbitrary ultrafilter 
Then, 
and 
see (5.3). By Proposition 12.4
(12.18)
By the closedness of 
and (12.16) 
So, from (12.18), we have the inclusion
Using (12.15), we obtain that 
Since the choice of 
was arbitrary, the inclusion
(12.19)
is established. Since the choice of 
was arbitrary too, from (12.17), we obtain that
So, the mapping 
is continuous at the point
. Since the choice of 
was arbitrary, the required inclusion (12.10) is established (see [16], (2.5.4)).
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
(12.20)
is defined; moreover, 
Proposition 12.6. The equality 
is valid.
Proof. Fix 
Then by (11.17) 
In addition, by (11.18) the obvious equality follows:
(12.21)
Moreover, by (9.24) we obtain that
(12.22)
Then, by Proposition 12.3 and (12.22) we have the equality chain
So,
. Since the choice of 
was arbitrary, 
Since (9.7) is valid, from Propositions 12.5 and 12.6, we have the important corollary connected with Proposition 5.2.1 of [9]:
(12.23)
is a compactificator, for which (in the considered case)
(12.24)
in (12.24) we use Proposition 3.1 and Corollary 3.1 of [18]. In addition, 
Therefore, by (12. 24)
(12.25)
In (12.25), we have the important particular case. We consider this case in the following section.
13. Ultrafilters as Generalized Solutions
We suppose that 
and 
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
(13.1)
is important. For this goal, we use the natural construction of Theorem 8.1 in [17]. Namely, we have the following Proposition 13. 1. If 
then 
Proof. Let 
Then, by the corresponding definition of Section 4 (see (4.3)) 
and, for some net 
in the set E,
(13.2)
Fix 
Then, by (13.2) 
Using (3.7), we choose 
such that 
(13.3)
Of course, 
And what is more, 
Indeed, let us assume the contrary:
(13.4)
Recall that 
Therefore, 
By (9.1) and (13.4) we have the inclusion 
Then, by (5.4) 
In particular (see (6.4)),
(13.5)
Moreover, by (5.3) 
Using (13.5), we obtain that
(13.6)
where 
From (13.6) and the second statement of (13.2) we have the following property: there exists 
such that 
(13.7)
By axioms of DS there exists 
for which 
and 
By (13.3) 
Moreover, by (13.7)
By (9.24) 
Therefore,
From (5.3), the inclusion 
follows; in particular, 
By (3.9) 
So,
We have the obvious contradiction. This contradiction means that (13.4) is impossible. So, 
Since the choice of 
was arbitrary, the inclusion 
is established. Then (see (9.26)), 
So, we obtain the inclusion
(13.8)
Choose arbitrary 
Then, by (9.26) 
and 
By Proposition 9.4 and [9,(3.3.7)], for some net 
in the set
, the convergence
(13.9)
is fulfilled. Now, we use axiom of choice. Fix 
Then, by the choice of 
the inclusion 
is fulfilled. Of course, by (5.4)
(13.10)
in addition, by (5.3)
. Since by (6.4) and (13.10) 
we have the inclusion
(13.11)
From (13.9) and (13.11), we have the property: for some 
we obtain that
.
(13.12)
From (9.24) and (13.12), we have the following property: 
(13.13)
By (5.3) and (13.13) we obtain that, for 
with the property 
the inclusion 
is valid and, as a corollary, by (3.9) 
So, 
and
Then, by (3.7) 
Since the choice of 
was arbitrary, the inclusion
(13.14)
is established. So, by (13.9) and (13.14) we obtain that the net 
in the set E has the following properties:
By definition of Section 4 (see (4.3)) 
So, the inclusion
is established. Using (13.8), we have the required equality
From (12.25) and Proposition 13.1, we have the following Theorem 13.1. If 
then, AS in 
with constraints of the asymptotic character defined by 
is realized by the rule
We note that, in Theorem 13.1, the set 
plays the role of the set of admissible generalized solutions.
14. 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 problems 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.
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