SOME REPRESENTATIONS CONNECTED WITH ULTRAFILTERS AND MAXIMAL LINKED SYSTEMS

Ultrafilters and maximal linked systems (MLS) of a lattice of sets are considered. Two following variants of topological equipment are investigated: the Stone and Wallman topologies. These two variants are used both in the case of ultrafilters and for space of MLS. Under Wallman equipment, an analog of superextension is realized. Namely, the space of MLS with topology of the Wallman type is supercompact topological space. By two above-mentioned equipments a bitopological space is realized.


Introduction
In connection with the supercompactness property, maximal linked systems (MLS) of closed sets in a topological space (TS) are investigated (see [1][2][3][4]; in particular, we note the important statement of [3] about supercompactness of metrizable compactums). The space of closed MLS with topology of the Wallman type is a superextension of the initial TS. Now, following [5], we consider more general approach. Namely, we suppose that a lattice of subsets of arbitrary nonempty set is given. Then, MLS of sets of this lattice are investigated. In particular, the lattice of closed sets in a TS can be used. Then, we obtain the above-mentioned variant of [1][2][3][4]. But, many other realizations are possible. For example, we can consider an algebra of sets as variant of the above-mentioned lattice. Note by the way, that in this case the Stone topology on the ultrafilter space is very natural. Since in many respects, MLS are similar to ultrafilters, the Stone equipment is submitted natural and for space of MLS. So, the idea of emploument of the two types of topologies arises: we keep in mind the Wallman and Stone variants.
We recall that ultrafilters were used as generalized elements in problems connected with attainability under constraints of asymptotic character (see, for example, [6][7][8]). Now, we seek to explore spaces which are comprehending for ultrafilters. In this article, it is established that the space of MLS is comprehending in this sense. In addition, it is logical to consider two characteristic types of topologies both for ultrafilters and for MLS. And what is more, we obtain two bitopological spaces (as a bitopological space, we consider every set equipped with two comparable topologies; in this connection, we note monograph [9]).
The case when two above-mentioned topologies coincide, we consider as degenerate. In the following, characteristic cases of such degeneracy are established (a variant of non-degenerate realization of bitopological space specified also). We indicate important types of lattices for which above-mentioned constructions are realized sufficiently understandably.

General notions and designations
We use the standard set-theoretical symbolics (quatifiers, connectives and so on); ∅ is anempty set and △ = is the equality by definition. We call a set by a family in the case when every element of this set is a set also. We take the axiom of choice.
For every objects x and y, we denote by {x; y} the set containing x and y and not containing no other elements. If h is an object, then we suppose that {h} △ = {h; h}. Of course, sets are objects.
Therefore, by [10, ch. II, §3], for every objects u and v, we suppose that (u, v) △ = {u}; {u; v} receiving the ordered pair with first element u and second element v. If z is an arbitrary ordered pair, then by pr 1 (z) and pr 2 (z) we denote the first and second elements of z respectively; of course, z = pr 1 (z), pr 2 (z) and pr 1 (z) and pr 2 (z) are defined uniquely.
If X is a set, then by P(X) we denote the family of all subsets of X and suppose that Fin(X) is the family of all finite nonempty subsets of X; of course. Fin(X) ⊂ P ′ (X), where P ′ (X) △ = P(X)\{∅} is the family of all nonempty subsets of X. In addition, a family can be used as X. For every nonempty family X, we suppose that (1. 2) In addition (see (1.2)), for any set S and a family S ∈ P ′ P(S) , the equality S = C S C S [S] is realized. If A is a nonempty family and B is a set, then is trace of A on the set B. Usually, in (1.3), the variant A ∈ P ′ P(A) and B ∈ P(A), where A is a set, is considered. For any sets A and B, by B A the set of all mappings from A into B is denoted. Under f ∈ B A and a ∈ A, by f (a), f (a) ∈ B, the value of f at the point a is denoted. For f ∈ B A and C ∈ P(A), Special families. In given item, we fix a set I (the case I = ∅ is not excluded). In the form of of all lattices of subsets of I with zero and unit . In addition, by the family of all algebras of subsets of I is defined. Moreover, by and (analogously) we define the families of all open and closed [11] topologies on I respectively. So, by (1.7)-(1.9) we obtain many useful examples of lattices of the family (1.6). Yet one particular case of a lattice of subsets of I is connected with σ-topologies of A.D. Alexandroff [12]: where as usually N △ = {1; 2; . . .}. Of course, under A ∈ (alg)[I], in the form of (I, A), we obtain a measurable space with algebra of sets. If τ ∈ (top)[I], then (I, τ ) is a topological space (TS). In addition, we use the notions T 1 -and T 2 -space (see [13,Ch.1]). Moreover, we use compactness [13,Ch.3] and other notions relating to general topology; see [13]. In particular, under τ ∈ (top)[I], by (τ −comp)[I] the family of all compact in (I, τ ) subsets of I is denoted; (τ −comp)[I] ∈ P ′ P(I) . We note the obvious property (1.10) Of course, in (1.10) we have an insignificant transformation of initial lattice.
Moreover, let Bases and subbases. For brevity of desingnations, until end of this section, we fix a nonempty set X and use (1.1). Then, is the family of all bases of TS (X, τ ). In addition, is the family of all open subbases of topologies on X. For any X ∈ (p − BAS)[X], we obtain that so, we obtain the family of all open subbases of TS (X, τ ). It is useful to introduce one auxiliary construction of [6]: moreover, it is logical to consider the following family: If τ ∈ (top)[X], then we suppose that so, we introduce the family of all closed bases of topologies on X. Of course, then, the family of all closed bases of TS (X, τ ) is defined. Now, we introduce the family of all closed subbases of topologies on X : Respectively, in the form we obtain the family of all closed subbases of TS (X, τ ). In addition, Recall following useful duality relations: We note also [6, (1.20)] and some simple corollaries of [6, (1.17)]: Now, consider some analogs concerning to subbases. In particular, (1.14) In terms of (1.14), we obtain the next analog of (1.12): As a corollary, from (1.15), it follows that ∀ τ ∈ (top)[X] (1.16) A special family of lattices. By (1.10) we can consider lattices from (LAT) [X]. Now, we introduce the family (1.17) It is possible to consider elements of (1.17) as lattices of small subsets of X. It is obvious that (1.18) The relation (1.18) assumes an amplification. For this, we introduce realizes the initial lattice L ∪ {X} as the lattice C X τ 0 L [X] of closed sets in T 1 -space: . Now, we consider some examples.
We consider the scheme of the proof of (1.24). In addition, we recall some known properties. So, at first, we show that by definition of the compactness property. Consider A ∩ B. By separability of (X, τ ) we have that then by transitivity of the operation of passage to a subspace of TS we have the equality and, as a corollary, . Since the choice of A and B was arbitrary, it is established that . Then, in particular, T ∈ P ′ P(X) and we have the set . Since the choice of T was arbitrary, we establish (see (1.30 . Since the choice of F was arbitrary, we obtain that So, the required property is known cofinite topology and is the family of closed sets in this topology.
Example 1.4. Let X, X = ∅, be uncountable set. Consider the family ω[X] of all no more than countable subsets of X. In addition, The corresponding proof is similar to previous example.
Coverings and linked families. Recall that X is a nonempty set. If X ∈ P ′ P(X) , then is the family of all coverings of X by sets from X . Let Then, the family of all linked systems of subsets of X is introduced. Moreover, suppose that We obtain the family of all MLS of subsets of X. In the following, we consider MLS containing in a given family. So, under X ∈ P ′ P(X) and by analogy with (1.34) In (1.36), we obtain the family of all MLS containing in the family X.
(1.37) P r o o f. Fix X and E with above-mentioned properties. In particular, X ∩ E ∈ P ′ P(X) . Let U ∈ X ∩ E and V ∈ X ∩ E. Then, in particular, U ∈ E and V ∈ E. By (1.33) we obtain that U ∩ V = ∅. Since the choice of U and V was arbitrary, we have the property Supercompactness. If τ ∈ (top)[X], then we suppose that 38) is the family of all closed binary subbases of TS (X, τ )); it is obvious that In addition, we suppose that , in the form of (X, τ ), we obtain (in particular) a compact TS.

Maximal linked systems and ultrafilters: general properties
In the following, a nonempty set E is fixed. We consider families from P ′ P(E) . In addition, we use (1.4)-(1.10).
Filters and ultrafiltres. In the following, we fix L ∈ π[E] (later, with respect to L, additional conditions will overlap). We consider (E, L) as widely understood measurable space. Then, is the family of all filters of (E, L). Maximal filters are called ultrafilters (u/f). Then is the nonempty family of all u/f of (E, L). If x ∈ E, then is trivial (fixed) filter corresponding to the point x. It is known [14, (5.9)] that Then, how easy check, follows. In addition, topology realizes [14] zero-dimensional T 2 -space Everywhere in the future, we suppose that (2.6) By (1.5) and (2.6) we obtain that Φ L (L 1 ∪ L 2 ) ∈ P F * 0 (L) is defined under L 1 ∈ L and L 2 ∈ L; in addition [6], . And what is more, in our case (under (2.6)) Remark 2.1. From (2.4) and (2.7), the following singularity is noticcable: for (UF)[E; L], properties of L are repeated. In this connection, we recall [6, (9.6)]: Returning to general case of (2.6), we note that (see [ In addition, topology (2.9) converts [6, Section 6] F * 0 (L) in a compact T 1 -space (2.10) We consider (2.5) as analog of Stone space and (2.10) as analog of Wallman space (the space of Wallman extension). In addition (see [15,Proposition 4.1]) With regard to (2.11), we consider triplet as a bitopological space (BTS); in this connection, see [9]. We do not discuss inessential differences with constructions of [9] and follow to above-mentioned interpretation of (2.12). So, (2.14) Moreover, easy to check that (we use the maximality property). With employment of the Zorn lemma, we obtain that Finally, we note the following corollary of maximality of MLS: Therefore, we obtain that The property (2.14) is complemented by the following equality:

Maximal linked systems; topology of the Wallman type
We recall that (2.14)-(2.18) are fulfilled under L = P(E) (the lattice of all subsets of E). In addition, (link) (1.34)). As variant of (1.36) and (2.15), we obtain that By (2.16) we obtain that Now, we return to arbitrary fixed lattice (2.6). Using (3.1), we consider one property of MLS for lattice (2.6). But, at first, we note one simple corollary of Proposition 1.
Proposition 2. The following property takes place:  Of course, we have the following particular cases: (here and later, we follow to [5]). Using (3.8) and (3.9), we obtain that The basic properties of the family (3.10) are considered later. Now, we pass to equipment by topology of the Wallman type. For this, we note that In the form of (3.11), we obtain the lattice dual with respect to L.
Remark 3.1 . We recall (see [5] . So, for the particular case, when (E, L) is a measurable space with algebra of sets, the dual lattice (3.11) coincides with L.
Under Λ ∈ C E [L], we suppose that of course, we can consider that Λ = E \ L, where L ∈ L. In this connection, we note that As a corollary, we obtain that by statements of Section 1 In addition, by (3.13) the following equality is realized: Of course, for (3.23), we use the property (indeed, (3.24) is obvious corollary of (3.18)). We consider (3.15) as a topology of Wallman type. We note two obvious property. Namely, for . Now, we consider the corresponding equipment for the set of u/f of the lattice L. For Λ ∈ C E [L], we obtain that So, we obtain the following property (see [5]): namely, In (3.27) and (3.28), we have analog of (3.15) and (3.17) respectively; in addition, it is useful to note that ∅ = F C [L| ∅] ∈ F C [L] (we use (3.11)) and therefore On the other hand, by (3.14), (3.25) and (3.26) From (3.29), we obtain the following statement of [5]: (2.10) is a subspace of TS (3.16). Namely As a corollary, we obtain the useful property: the set F * 0 (L) is compact in TS (3.16): Now, the following statement is obvious.
is closed in this space: In connection with Proposition 4, we note the known property concerning to [4, 4.16] (see too [16, p. 65]).
We note that, for every E ∈ (L − link) 0 [E], the following equality is realized: as a corollary, by (3.19) we obtain that So, we have the following statement of [5].
Of course, if (3.16) is a T 2 -space, then it is a supercompactum. We note that by the maximality

Maximal linked systems as elements of zero-dimensional T 2 -space and bitopological structure
In this section, we introduce TS analogous to (2.5). Elements of this new TS are MLS. We recall that by (1.14), (3.8), (3.10), and (3.9) So, by (4.2) we obtain the required TS For this TS, by (4.2) we have the inclusion We obtain some analog of (2.13). So, the family C * 0 [E; L] serves both topology T * (E| L) and topology T 0 (E| L). But, now we focus on consideration of TS (4.3).

It is easy proved that
(4.6) in (4.5), we use the property analogous to (3.3). We use (4.6) for verification of separability of the TS (4.2). For this, we introduce the next notion: if E 1 and E 2 are nonempty families, then Of course, in (4.7), we can use arbitrary MLS from (L − link) 0 [E] as E 1 and E 2 . Then, by (4.6) and (4.7) (4.8) We confine ourselves to employment of open neighborhoods. Of course, by (3.10) and (4.2) we obtain the following obvious property: if We note that by (3.32), for is realized (see (4.7)). As a corollary, by (4.8) and (4. On the other hand, from (4.11) the following property (see [5]) is extracted: From (3.10) and (4.12), we obtain that Using axioms of TS, from (4.13), we obtain that The corresponding proof (see [5]) is immediate combination of (4.10), (4.14), and (4.15) (see [13, 6.2]). We note the following obvious property (see (3.8) and definitions of Section 2) Therefore, by (2.4), (3.10), and (4.16) we obtain the equality . As a corollary, the following important property (see [5]) is realized:
Corollary 1. If L 1 ∈ L and L 2 ∈ L, then The corresponding proof is obvious (see Proposition 7). So, mapping is a bijection from L onto C * 0 [E; L] (see (3.10) and Corollary 1). We note that from (5.6) the next density property follows:

Some additions
In this sections, at first, we consider questions meaningful of a duality for families C * 0 [E; L] and C 0 op [E; L]. For this, we recall that (see Section 3) As a corollary, by (4.4) and (6.1) we have the property Then, by (1.16) we obtain (6.3).
From (3.17) and Proposition 8 we have the following property In (6.2) and (6.4), we obtain a duality of subbases.

Bitopological space of closed ultrafilters and maximal linked systems
We recall (5.6). Then, by this property the topologies T 0 L [E] and T * L [E] are similar (later, we show that in many cases the above-mentioned topologies are equal). But, now we consider the variant of the set lattice for which the above-mentioned topologies differ typically. Namely, we fix τ ∈ (D − top)[E]; so, τ ∈ (top)[E] for which (E, τ ) is a T 1 -space and (in this section) we suppose that Of course, by (7.1) and (7.2) we can use statements of Section 5. In particular, by (5.6), (7.1), and (7.2) At the same time, we have (see [5, § 7]) the property So, by (7.5) the following statement is realized: TS (2.10) feels subsets of E accurate to closure. We recall [5, (7.3)]: for A ∈ P(E) and With employment of (7.6), we obtain (see [5, (7.4)]) in our case (7.7) Finally, by [5, Theorem 7.1] we obtain the following implication: So, for (7.1) and nondiscrete T 1 -space (E, τ ), BTS (2.12) is nondegenerate. From (4.18) and (7.8), we obtain that We use (7.8) and (7.9) in connection with lattices of the family (1.17).

Some particular cases
In this section, we fix a lattice L ∈ (↓ −LAT) 0 [E].
In addition, in the form of the triplet   under L ∈ L. We recall that, under Λ ∈ L, the inclusion E \ Λ ∈ L is realized; and what is more, by (3.13) As a simple corollary, in our case, the equality is realized. Therefore, by (3.15) and (4.2) T 0 (E| L) = T * (E| L). is a nonempty supercompactum. In particular, (9.6) is a nonempty compactum. The corresponding proof follows from Proposition 4 (indeed, for (9.6) we have the separability property). In connection with Proposition 9, we recall (3.31).

Conclusion
We reviewed two BTS. In the first case point of BTS are MLS and, in the second case, similar points are u/f of a set lattice. It is established that the second BTS can be considered as a subspace of the first BTS. We indicated the natural variants of our lattice for which the above-mentioned BTS are degenerate and, opposite, the variants with degeneracy of the corresponding BTS is absent. Our consideration is connected with ideas of supercompactness and superextension of a TS. For degenerate BTS the corresponding space of MLS is a supercompactum. Under consideration of the lattice of closed MLS, we obtain a non-degenerate BTS typically.