1. Informative Discussion of Problem
We consider general attainability problem in a topological space (TS). For this, we fix a nonempty set E elements of which play the role of usual solutions (sometimes, it is logical to consider elements of E as controls). Moreover, we fix a TS and a mapping called the goal operator. If , then is the concrete result of the choice of e. If a subset of E is given, then image of can be considered as attainable set or analog of attainability domain in control theory (see   ). The closure of is a natural generalization: we assume realization of points of “in a limit” under precise validity of constraints connected with the set . In investigation of J. Warga, for optimization problems, the approach using weakening of -constraints was proposed (see ch. III,IV of monograph  ). Of course, this approach assumes the natural spreading on attainability problems. Namely, we replace by a nonempty family of subsets of E with the property
(usually, the family is directed in the following traditional sense
but, now we not discuss this supposition). It is logical to suppose that sets of the family are “near” to (we keep in mind Warga approach). For every set , we consider the set-image and its closure in TS . Later, we can consider the intersection of all sets instead of ; in this representation, the validity of (1.2) is essential. The given intersection is interpreted as attraction set (AS). This construction can be considered as natural analog of the Warga approach for extremal problems (see ch. III, IV in  ). But, for abstract attainability problem, the above-mentioned construction with employment of AS can be extended very essentially.
Indeed, we can investigate the setting for which the family is primarily (it is supposed that the set is not given). This family consists of subsets of E; usually it is sufficient to suppose that is directed (see (1.2)). But, now we not require validity of this condition.
So, let be a nonempty family of subsets of E. At first, we consider questions connected with the choice of mappings with values in E. In particular, we can consider the choice of sequences in E. But, sometimes, our possibilities are extended in the case when the choice of nets is admissible. Therefore, now we discuss the choice of nets ( ) in the set E. We consider such nets as asymptotic regimes. Respectively, in , nets are realized. If, for a net ( ) in E, under every , the condition is fulfilled starting from a certain index, then we consider ( ) as admissible asymptotic regime. The following question arises: for which , an admissible asymptotic regime ( ) with the property of convergence of to y can be chosen. Such points y are interpreted as attainable elements of . We consider the problem about construction of the set of all attainable elements. We call his AS. In addition, under condition (1.2) our new definition of AS is equivalent to previously mentioned definition (see formula (3.3.10) in  ). If (1.2) is not fufilled, then we can to replace on the family of all finite intersections of sets of . Then AS defined by the convergence of nets (that is, by asymptotic regimes) coincides with intersection of all sets . We note that filters can be used instead of nets (see  ).
2. General Notions and Definitions
We exploit standard set-theoretical symbolics (quantors, propositional connec- tives; is empty set). In the following, is equality by definition; replaces phrase “exists and uniquely”. We take axiom of choice. For every object x, by we denote singleton containing x (so, ). The set is called a family in the case when every element of this set is a set also.
If H is a set, then by and we denote the families of all and all nonempty subsets of H respectively; so, . By we denote the family of all finite sets of ; so, is the family of all nonempty finite subsets of H. If , then is a nonempty subfamily of . Then, for a fixed set and a family , we introduce the family
dual with respect to (of course, a topology of can be used as ; in this case, the family of closed subsets of is realized as ). It is obviously that
Now, we consider other operations with families. So, for every nonempty family and a set Y,
is trace of on the set Y. If is a nonempty family, then we suppose that
in addition, and . Of course, in (2.1), we obtain two nonempty families of subsets of union of all sets of .
For every sets A and B, by we denote the set of all mappings from A into B; of course, under and , is the f-image of the point a. For every sets U and V, a mapping , and a set , by we denote the contraction of g on the set W ( and ) and (the image of W under operation by g).
Special families. In the following, we consider families of subsets of that or different fixed set. Therefore, in this item, we fix a nonempty set I and consider the family of all nonempty subfamilies of . Among all such subfamilies, we select p-systems (see p. 14 in monograph  ) with “zero” and “unit”; then,
is the family of all such p-systems. Of course, every algebra of subsets of I and every topology on I are p-systems of the family (2.2); therefore, we introduce
Every semialgebra of subsets of I (semiring with “unit”) is a p-system of (2.2) also. In the form of
we obtain the family of all separable π-systems of (2.2). Particular case of π- system of the family (2.3) is realized for p-systems with singletons:
(in (2.4), we exploit the obvious property ). In the form of
we obtain the family of all lattices with “zero” and “unit”. Of course, and . We suppose that
(the family of all nonempty directed subfamilies of ); then, elements of the family
are filter bases of the set I and only they. Moreover,
is the family of all filters of I (see ch.I in monograph  ); elements of the family
are ultrafilters of I and only they. If , then we suppose that
In addition, the following known (see ch.I in  ) construction is used: if , then
is the filter of I generated by the base . Finally, for
so, trivial (fixed) ultrafilters are defined. By the natural rule of immersion of into is realized. We note that (2.6) is a nonempty set.
Filters and ultrafilters of p-systems. Now, we consider “partial” filters and ultrafilters. For this, in the present item, we fix arbitrary p-system . So, is analog of measurable space. In the following, we call a meaurable space also; of course, this term is regarded extendedly. We suppose that
Families of (2.8) are nonempty centered subfamilies of and only they. Introduce “partial” analogs of filters of (2.5); namely, in
the set of all -filters is defined. As a corollary,
is the set of all -ultrafilters. In connection with (2.10), we note constructions of   . Suppose that
The sets (2.11) play important role in questions of topological equipment of ; these questions is considered later. On the other hand,
in (2.12), we obtain the natural analog of (2.7). Now, we consider analogs of (2.11) for which the condition of -measurability of set defifning (2.11) can be omitted. Namely, let
Analogously, as some generalization of (2.12), we exploit the following families:
Of course, under ; moreover, under . So, (2.13) is natural generalization of (2.11) and (2.14) is analogous generalization of (2.12). If , then
is considered as trivial -filter corresponding to point x. We recall (see Section 3 in  ) that
where In addition (see p.214 in  ),
Remark 2.1. We recall that (and what is more, ) and consider the particular case . From (2.5) and (2.9), the equality follows. As a corollary, by (2.6) and (2.10) . Using (2.7) and (2.12), we obtain that Of course, under . ,
Example. Let be real line. In this item, we fix , and suppose that . Let
The family (2.17) is a semialgebra of subsets of I. In this item, we suppose that is algebra of subsets of generated by semialgebra and
So, here is a measurable space with algebra of sets. Then, by statements of 
We note that (see (2.15) and constructions of Section 6 in  ) in our case
since (in our case) , by (2.16) we obtain that Therefore, by (2.15) and (2.20) in our case
In (2.21), we obtain exhausting representation of for considered variant of .
3. Elements of Topology
For every topology , where is a nonempty set, we obtain TS ; by the family of all subsets of compact in TS is denoted. Of course, . And what is more, . In addition, for every point
and, in particular, ; therefore,
is the filter of all neighborhoods (see definition of ch.I in monograph  ) of x in TS . If , then
(the closure of A is introduced); moreover, we sequentially introduce interior and frontier of A:
Finally, and are the family of all subsets of every dense (in ) and the set of all isolated points of respectively. In correspondence with definition of ch.I of  , we suppose that
By (3.1) the convergence of filter bases is defined (we note that ; moreover, under ). We recall that, under , topology converts in subspace of TS .
We introduce the families and of all canonically open and closed sets res-pectively.
If , and , are TS, then
is the set of all continuous mappings from into ; recall that, for and , the property is realized. We note that for every and a family , the (nonempty) family
is defined. Moreover, for and , we obtain that .
Now, we consider some topological properties of ultrafilter space for a fixed p-system. So, we fix until end of this section a nonempty set I and a π-system . Using (2.11), in the form of
we obtain the base of a topology of . This topology is defined by the rule
In addition, under , the property
is realized (this is the simple corollary of (2.11)). In the form of
a zero-dimensional Hausdorff TS is realized.
Remark 3.1. If , then (3.3) is a compactum (compact Hausdorff TS). In this case, (3.3) is the Stone space. For this case,
So, we obtain (see (3.4)) the important particular case of (3.2), (3.3). ,
Until end of this section, we suppose that . By (2.16) we obtain the mapping
denoted as . In addition, . Now, we recall some statements of  connected with (3.5). Of course, under , in the form of
we obtain the image of S under operation (3.5). Then (see  )
As a corollary, Finally,
So, We recall that (see (2.3)). Therefore, (3.6) and (3.7) are fulfilled under .
4. Attainability under Constraints of Asymptotic Character
Returning to Section 1, we preserve the designation E for the space of usual solutions. Families are considered as constraints of asymptotic character (see Section 1). Moreover, in this section, we fix a TS and a mapping . We discuss construction of AS in . These AS can be used as basic or as auxiliary. In Section 1, nets in E were used as distinctive “asymptotic solurions”. But, now the analogous employment of filters and ultrafilters is more appropriate. The equivalence of the representations with employment of nets and ultrafilters was noted (in particular) in §3 of  . Now, we discuss the representation realized in class of ultrafilters. So, by definition of §3 of 
We note that the following particular case is useful:
In connection with (4.2), we recall discussion in Section 1 (see (1.2)). We consider the sets (4.1) and (4.2) as AS corresponding to given nonempty family of subsets of E (we recall that, under employment of nets in E, we obtain the equivalent representation of AS; in this connection, see constructions of §3 in  ). By (4.1) the following singularity is realized: ultrafilters play the role of approximate solutions of J.Warga.
5. Representations of Attraction Sets
In this section, we consider some transformations of AS. In addition, we fix a TS and (see Section 1). Under , we consi- der
as the basic AS. Namely, in the case when defines constraints of asymptotic character, (5.1) is our goal set. For investigation of the set (5.1) we can exploit auxiliary AS. We note the following general property: if is a TS, and , then
This property is established very simply with employment of continuity of the mapping g and (4.1). From relations (2.8.1), (2.8.2), and Proposition 3.3.1 of  , under conditions with respect to used in (2), we obtain that
of course, we use (4.1). We recall that the continuous mapping g satisfying to two first conditions in premise of the implication (5.3) is called almost perfect; see Section 3.7 of monograph  . In connection with (5.3), we note the important particular case of such mapping (see formula (2.8.7) of  ): if is a compact TS and is a Hausdorff space, then
Therefore, the natural compactified case arises. In this connection, in the folowing, we suppose always that
is a Hausdorff TS. The case of such spaces (5.5) is sufficient for majority appli- cations. Then (in our case of TS (5.5)) for every compact TS and , the implication
In connection with the given important property (5.6), we introduce the special notion of a compactifier. Namely  , we call compactifier every four such that is a compact TS, , , and . From (5.6), we obtain that for every compactifier
So, every compactifier realizes the representation of the basic AS as continuous image of auxiliary AS in compact space. In connection with (5.7), we note that (closed) set
has the following obvious property (see (4.1)):
moreover, under , from (5.8), we obtain that
where (here and later) the designation
is used. So, by (4.1) and (5.9) we obtain that, for Hausdorff TS
(subspace of the initial TS (5.5)), the following equalities
are realized; of course, we use the next obvious property: (see (5.8)). In connection with (5.12), we suppose that (by (5.10)) the replacement (5.5) ® (5.11) is realized.
Now, we consider the corresponding reduction of compactifiers defined in terms of TS (5.5).
Proposition 5.1. If is a compactifier, then .
Proof. Indeed, we obtain that
where (5.4) is used (really, is a closed set). ,
Proposition 5.2. If is a compactifier, then
is compactifier also.
The corresponding proof is given in Section 4 of  .
In the following, we call a compactifier dense in the case when . By Proposition 5.2 we obtain that (as in Proposition 4.1 of  ) every compactifier assumes transformation to a dense compactifier. Therefore, later, we consider only dense compactifiers. It is obvious the following
Proposition 5.3. If is a dense compactifier, then is a surjection from onto .
Proof. Indeed, by (5.4) we obtain that
(in (5.13) we use constructions of  ; in particular, see relations (2.8.1)- (2.8.4) and (2.8.7) of  ). ,
Corollary 5.1. If is a dense compactifier for which g is an injective mapping from Y in , then g is a homeomorphism from TS onto TS (5.11).
Proof. Let all conditions with respect to are fulfilled. Then, by Proposition 5.3 the mapping g is a bijection from Y onto . Moreover, let . Using (5.10), we choose for which . Then
(we recall the obvious property: for the inclusion is realized; see Proposition 5.3). Since the choice of was arbitrary, from (5.14), the property follows. But, is compact and (5.11) is Hausdorff TS. Therefore, g is a homeomorphism from onto (5.11) (indeed, g is a continuous bijection from compact TS onto Hausdorff TS; now, we use Theorem 3.1.13 of monograph  ).
6. Compactifiers and Quotient Topologies
We recall the known notion: if is a TS, Y is a nonempty set, and , then
is called quotient topology on Y (this topology was introduced by P. S. Alexandroff and H. Hopf). In supplement to (6.1), we note the obvious property: for every nonempty sets X and Y, a surjection g from X onto Y, and a set , the equality
is fulfilled ((6.2) is known property of surjections).
Remark 6.1. Let be TS, Y be a nonempty set, and . Moreover, we suppose that is topology (6.1). Then,
For more brief presentation, we introduce some new designations. So, for every nonempty sets X and Y, by we denote the set of all surjections from X onto Y. If and are two TS, then
In (6.3) and (6.4), closed and open continuous mappings are introduced. In connection with (6.3) and (6.4), we note that (see ch.3 of monograph  ) for every TS and , the following property takes place: if , then
As a simple corollary, the following property is realized: if is a compact TS, is a Hausdorff TS, and , then
We note that conditions realizing (6.5) correspond to variant used under employment of dense compactifiers (see Proposition 5.3).
Proposition 6.1. If is a dense compactifier, then
Proof. From Proposition 5.3 and constructions similar to (5.14), we obtain that
where is a compact and is a Hausdorff TS. So, (6) follows from (6.5). ,
We obtain that the closed subspace of corresponding to (5.8) is TS with quotient topology with respect to every dense compactifier. We note two obvious (but useful) general statements.
Remark 6.2. Let and are TS. Moreover, let
Then, the equality is realized. We consider the brief scheme of the proof.
The inclusion follows from (6.3) and (6.7). Let . By (6.2) and (6.7) , where . So, . Since the choice of was arbitrary, we obtain the inclusion . ,
Remark 6.3. Let and correspond to Remark 6.2. Moreover, let
Then, . Indeed, by (6.4) and (6.8) . Let . Then, by (6.2) and (6.8)
where . Therefore, . The inclusion is esta- blished. ,
Proposition 6.2. If is a dense compactifier, then
Proof. By Proposition 5.3 . In addition, similarly to (5.14), we obtain that . Since is compact and is a Hausdorff TS, ; see (2.8.2) and (2.8.7) in monograph  . From Remark 6.2, the equality (6.9) follows. ,
Proposiotion 6.3. If is a dense compactifier, then
Proof. We recall that (see discussion in connection with (5.14)). Then, . Therefore,
Let . Then, (indeed, (5.11) is a Hausdorff TS). By Proposition 6.2 , where since is a compact TS. Then, . So, the inclusion
is realized. Using (6.11), we obtain (6.10). ,
Remark 6.4. We note some properties connected with Proposition 6.1. Let be a fixed dense compactifier. Then
Indeed, let . Since by Proposition 5.3, we obtain that , where by continuity of g (see (5.14)). As a result . Since the choice of was arbitrary, the required inclusion (6.12) is realized. The analogous construction realizes the inclusion
where (6.2) and continuity of g are used. Indeed, for , we obtain that (continuity of g is used) and by (6.2). So, . The inclusion (6.13) is established. ,
7. Some Transformations of Compactifiers
In this section, we fix a dense compactifier . Then
(we use reasons similar to (5.14)). Now, we consider the natural transforma- tion of in injective compactifier (see Proposition 5.3). In addition, constructions of  are used. The family
is a partition of Y; of course, (7.2) generates the following equivalence relation º on Y by the rule:
In addition, . We note that . Therefore, we introduce the mapping by the next rule:
From (7.1) and (7.4), we obtain that . And what is more, is a bijection from onto .
Now, we obtain that . So, the canonical projection from Y onto the corresponding factor-space is defined. In addition,
(the canonical expansion of g). Indeed, let and . In addition, ;
Since , by (7.4) we obtain that . So, by (7.6) . Since the choice of was arbitrary, the property
is realized. As a result, we obtain (7.5).
Following constructions of Section 2.4 of monograph  , we equip (see  ) the set with quotient topology. Namely,
So, we obtain TS for which
Since , the following property is realized: is a compact TS (we apply the continuity of realized in (7.8) and compactness of ).
Now, we note that (by the choice of ) and
Finally, . Indeed, let . By (7.1) we obtain that . By (7.5)
where . Since , by (7.7) . Since the choice of was arbitrary also, the required continuity property is established. As a result,
In particular, is a compactifier (indeed, by (7.10) the obvious property follows). And what is more, is a continuous bijection from onto . Since is a compact space and is a Hausdorff space, the mapping is a homeomorphism from onto . Then, is an injective compactifier. We note the density property of this compactifier.
Indeed, by the choice of the equality . Then, by (7.8)
So, . We obtain that is an injective dense compactifier.
In addition, is a Hausdorff TS. Of course, this property follows from homeomorphism of and separability of . So, is compactum. Moreover, and, as a corollary, TS is a compactum also.
Remark 7.1. We note the last property. Indeed, if (5.5) is a Hausdorff TS for which a compactifier exists, then (5.11) is a compactum. Now, we consider the scheme of the corresponding proof.
So, as previously, we consider Hausdorff space (5.5). Therefore, (5.11) is a Hausdorff space also. We consider the case when some compactifier exists. By Proposition 5.2 we obtain that
is a dense compactifier. Now, we suppose that
By (7.12) we rename compactifier (7.11). So, is our dense com- pactifier. Later, we apply procedure of the present section; as a result, we obtain that is a compactum.
8. Extension in Class of Ultrafilters of π-Systems
Now, we consider a special variant of general construction of Sections 5-7. But, at first, we introduce some new notions (see         ). We fix and consider the set
(we apply separability of TS (5.5); all mappings-constants with values in (and defined on ) are elements of the set (8.1)). From (8.1), the next definition follows; namely, if , then
is defined by the following condition: under , the point has the property
This corresponds to similar definition of §5 in  . Of course, by (8.3)
In connection with (4), we note statements in formula (5.4) of  ; namely,
So, in (8.5), we obtain analog of (5.2). We recall that TS is called a regular space (regular TS) in the case when is -space for which closed (in ) neighborhoods of every point realize in totality a local base of for this point x. In this connection, we recall Formula (5.5) of  : if is a regular TS, then
Until end of this section, we suppose that is a regular TS and . Moreover, now, we suppose that
Then, , and we obtain the natural concrete variant of (8.3)-(8.5); in addition, for
As an addition to (8.5), we note (see (8.6)) the property noted in Proposition 4 of  :
(we exploit (4.2), (5.2), (8.6), and (8.7)). As a corollary, from (8.9), we obtain that
(we apply (4.1) and (4.2); indeed, under , the property follows). It is useful to apply Remark 3 of  .
In connection with (5.2), (8.7), and (8.10), we recall Proposition 3 of  :
Attraction sets and compactifiers. We recall that the case of regular space and is considered (in this case TS (5.11) is regular also). Moreover, in our case (8.7) is fulfilled. Finally, we suppose until end of this section that TS
is compact. Then TS (8.12) is a zero-dimensional compactum. Therefore, from (3.7), (8.6), and (8.10), we obtain that
is a dense compactifier. As a corollary, by (5.7) and (8.11)
In connection with (8.14), we note the following important particular case:
Of course, (8.15) is sufficient for majority of applications. From (8.4), (8.7), and (8.15), we obtain that
It is useful to note that by (8.15) and (8.16) (in our case) ultrafilters from , where , can be considered as analogs both generalized and approximate solutions of J. Warga (see constructions of ch.III of  ).
Remark 8.1. The compactness property of TS (8.12) is very important (see (8.14)-(8.16)). Now, we note only Remark 3.1 (of course, more general cases are known: see §6 in  ). Another important question is connected with (8.7). More exactly, we obtain the serious question about representation of the set (8.1). Now, we discuss only two possibilities for .
1) Let (5.11) be a compactum (in this connection, we note Remark 7.1). In addition, by (5.8) . We strive to obtain conditions sufficient for (8.7). For this, we apply the construction of §5 in  . For any , we introduce the family
of all local bases of TS at the point h. In this item, we suppose that
So, (8.17) is a condition of the “local measurability” of . We recall about compactness of TS (5.11). Then, by Corollary 5.1 of 
Let and be a point such that (see (8.18))
In addition, . By (3.1) and (8.19)
We recall that (see Formula (2.3.8) of  ) by (5.10)
We recall that
In addition, by the choice of we obtain that . So, the filter
is defined; by (8.22) the inclusion
follows. By (8.20) and (8.23) the next inclusion
is realized. Now , we exploit (2.5) and (8.21). Indeed, let . Then, by (8.21) . By (8.24) the inclusion follows, where . Since , by (2.5) . So, we obtain that . As a corollary,
where . In particular, . Since the choice of was arbitrary, it is established that
From (8.1), we obtain that, in our case, (8.7) is fulfilled. So, (8.17) and compactness of TS (5.11) are sufficient for (8.7).
2) Now, we consider the variant connected with application of uniform limits of real-valued step-functions (the more general construction is given in  ). In this item, we suppose that , where is a nonempty set. So, elements of are functionals on and only they. We apply and suppose that
In (8.25), we obtain real-valued components of . We recall that E is equipped with . Now, we suppose that is a semialgebra of subsets of E (of course, the variant of an algebra of subsets of E is possible also). Under , by we denote indicator of the set under , and under . By we denote the linear span of the set (we apply the pointwise linear operations in ) obtaining a subset of the set of all bounded real-valued functions on E. We equip the linear space with traditional sup-norm (see ch.IV, Section 2, item 13 of  ). Then, . The closure of in with topology generated by norm is denoted by (if , then corresponds to ch.IV, Section 5 of  ). In particular, with norm induced from is a Banach space.
Suppose that . Now, we define a variant of topology . Namely, we suppose that is the natural topology of the Tychonoff power of with usual -topology for the case when is the indexed set. So, we consider the Tichonoff product of samples under indexed set . Using Proposition 6.3 of  , we obtain that (8.7) is fulfilled (in  , the concrete variant of the mapping is indicated; also, we note Remark 6.2 of  ).
So, under some natural conditions, the property (8.7) is realized.
9. Some Topological Properties of Attraction Sets in Space of Ultrafilters of π-Systems
In this section, we fix and consider the set (see Section 2). In this construction, we follow to  .
Proposition 9.1. If and , then
in addition, if , then
The corresponding proof is given in §4 of article  . In connection with Proposition 9.1, we suppose until end of this section that . Then, by Proposition 9.1
As a corollary, in our case, the mapping
is a bijection from onto .
We fix until end of this section (we apply as the family defining constraints of asymptotic character; in particular, . In this and next sections, it is supposed that
(in §6 of  , the examples of and with (9.2) are given; we note that (9.2) is fulfilled under
the last case is considered in  specially). By (2.11) and (9.2) the set
is defined. We recall that coincides with intersection of all sets . In Theorem 6.1 of  , it is established that
In particular, from (9.3) and (9.4), the following property follows: the interior of is an open-closed set.
Remark 9.1. Recall (8.11). Then, under (for which and (9.2) is fulfilled), we obtain that
So, in this case, by (9.4) the interior of auxiliary AS is defined (see definitions of Section 2). Since , we obtain that
(the frontier of the set (9.5) is realized). By (9.6) the next inclusion is realized
Now, we consider questions connected with approximate realization of ultrafilters of . So, we consider questions connected with approximate realizability of elements of (9.6). Then
So, ultrafilters of the frontier of are not realizable “outwards”. In addition, by (3.6) we obtain that
Moreover, by (9.6) we obtain the following
Proposition 9.1. The following inclusion takes place:
Proof. By (3.6), (3.7), and additivity of the closure we obtain that
Since , by (9.6) we obtain the required inclusion (9.9). ,
Corollary 9.1. The next inclusion is realized:
The corresponding proof is reduced to immediate combination of (9.7), (9.8), and Proposition 9.1. So, ultrafilters of the frontier of assume approximate realization only by points of the set .
We recall that by (9.2) and (9.4)
Remark 9.2. If , then
Indeed, let . By (9.10)
Using (9.4) and (9.6), we obtain (under our condition) that
So, . ,
10. Some Additions
We follow to constructions of Section 4 of article  under consideration of the case , where E is a nonempty set. In this case, the open-closed sets are defined. In addition, under
As a corollary, we obtain the following equality
therefore (see Section 3) we obtain that
So, in our case, TS of type (3.3) has everywhere dense subset consisting from isolated points. These points are trivial ultrafilters.
From (10.1) and (10.2), we obtain ( under ) that
From (3.6) and (10.3), the following important property (see p. 222 in  ) is realized:
Until end of this section, we fix a family .
Proposition 10.1. Let (9.2) holds and . Then
Proof. By (9.6) we have that . Then, from Remark 9.2, we obtain the property
By (10.4) we obtain the required statement. ,
As a corollary, the following property is realized: if the conditions of Proposition 10.1 are fulfilled, then by (3.6)
Now, we discuss (10.5). The expression on the right-hand side of (10.5) corresponds to “standard” realization of generalized elements. Namely, we have some set of usual solutions. These solutions are transformed in generalized elements (here, in ultrafilters). The obtained set of ultrafilters is transformed to a closure (we obtain ). By (10.5) and Proposition 10.1 we no obtain under every (of course, we keep in mind that (9.2) and conditions of Proposition 10.1 are fulfilled and the sets-closures on the right-hand side of (10.5) are used). So, auxiliary AS (see (8.11) and Section 2) one cannot to obtain by closure of some subset of the initial set E.
This work was supported by Russian Foundation for Basic Research (project no.15-01-07909).