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33URAL MATHEMATICAL JOURNAL, Vol. 5, No. 1, 2019, pp. 31-47
DOI: 10.15826/umj.2019.1.004
TO A QUESTION ON THE SUPERCOMPACTNESS OF ULTRAFILTER SPACES1
Alexander G. Chentsov
Krasovskii Institute of Mathematics and Mechanics,
Ural Branch of the Russian Academy of Sciences, 16 S. Kovalevskaya str., Ekaterinburg, Russia, 620990 [email protected]
Abstract: The space of ultrafilters of a n-system endowed with the topology of Wallman type is considered. The question on the supercompactness of this space is investigated. For this, the enveloping space of maximal linked systems with the corresponding topology of Wallman type is used. Necessary and sufficient conditions for the coincidence of the set of all ultrafilters of the initial n-system and the set of all maximal linked systems for this n-system are obtained. Specific variants of wide sense measurable spaces with this coincidence property are given.
Keywords: Maximal linked system, Topology, Supercompactness, Ultrafilter.
Introduction
In the present paper, we study ultrafilters of wide sense measurable spaces. Every such measurable space is defined by introducing a n-system of subsets of a nonempty set. More precisely, a n-system is a family of sets closed with respect to finite intersections; see [1, p. 14]. We consider only n-systems with "zero" and "unit," i.e., the empty set and the enveloping set are elements of our n-systems. This approach enables the precise description of the ultrafilter set for a given measurable space (in this connection, see [2, 3]). The ultrafilter set can be equipped with a Wall-man type topology. We consider the equipping of this type as a basic thing. In addition, we use a Stone type topology. As a result, a bitopological space is realized. This bitopological space can be considered as a kind of a subspace of the bitopological space of maximal linked systems with Wallman and Stone equippings.
Note that Wallman and Stone type topologies are used in the theory of extension of topological spaces (see [4; 5, Sect. 3.6] and others). Ultrafilters play an important role in general topology. Constructions involving maximal linked systems are used in connection with the notions of topological space; in addition, usually, maximal linked systems of the family of closed sets in a T1-space are considered (see [4, Sect. 5.11], [6-8], and [9, Ch. VII, Sect. 4]). We emphasize the important result of [8] on the supercompactness of metrizable compact sets.
In [10], for superextension constructions, a natural bitopological equipping was realized. In the author's consequent papers [11-14], this approach of [10] was extended to the case when a n-system is used as an anticipating measurable structure (the approach of [6-8] corresponds to the case when this n-system is the lattice of closed sets in a T1 -space). Thus, in the above-mentioned general case of an arbitrary n-system, bitopological spaces of ultrafilters and maximal linked systems were constructed (see [11-14]). In addition, in the case of maximal linked systems, a topology of Wallman type generates a supercompact space.
1This work was supported by Russian Foundation for Basic Research (project no.18-01-00410).
Recall that ultrafilters are maximal linked systems. But often there are maximal linked systems that are not ultrafilters. The simplest example of such maximal linked system was given in [9, 4.18]. This example plays an important role in questions of representation of maximal linked systems.
At the same time, for some (widely understood) measurable space, all maximal linked systems are ultrafilters; in this connection, see [15, 16]. In the present paper, we investigate this case in more details. It can be interesting because the question on the supercompactness of the ultrafilter space with a Wallman type topology arises. In addition, the supercompactness property is not, generally speaking, hereditary (see [4, Sect. 5.11]). However, in the case when all maximal linked systems are ultrafilters, the required supercompactness property of the ultrafilter space can be obtained from the similar property of the space of maximal linked systems. This approach reduces to an interesting representation, but requires finding conditions for the above-mentioned coincidence of the ultrafilter set and the set of maximal linked systems. The present paper is devoted to this issue.
1. General notions and notation
We use the standard set-theoretic symbolism (quantifiers and connectives); △ stands for the equality by definition and 0 for the empty set. In what follows, a family is a set whose elements are also sets. We accept the axiom of choice. For any objects x and y, we denote by {x; y} the set containing x and y (as proper elements) and not containing other elements; thus, {x; y} is an
unordered pair of the objects x and y. If z is an object, then {z} △ {z; z} is a singleton containing
z as a proper element. For any three objects u, v, and w, we set {u; v; w} △ {u; v} U {w}. For
any objects p and q, (p, q) △ {{p}; {p; q}} is the ordered pair [17, Ch. II, Sect. 3] with the first element p and the second element q. If z is an ordered pair, then pr1 (z) and pr2(z) are the first and the second elements of z, respectively.
If X is a set, then P(X) is, by definition, the family of all subsets of X, P'(X) = P(X) \ {0}, and Fin(X) is the family of all finite sets of P'(X); thus, Fin(X) is the family of all nonempty finite subsets of X. Certainly, a family can be used as X. In this connection, we introduce the following notation [12, Sect. 2] for a fixed nonempty family X:
{U}(X) △ { J X : X € P(X)}, {n}(X) △ { p| X : X € P'(X)},
X ex xex
{U}„(X) △ { J X : K € Fin(X)}, {n}„(X) △ { f| X : K € Fin(X)}.
xeK XeK
If M is a set and M € P'(P(M)), then
Cm[m] △ {M \ M : M € M} € P'(P(M)) is the family dual to M. For any set H and a family H € P'(P(H)), we have
H = Ch [CH [H]]. For any nonempty family S and a set L, define
{S n L : S € S} € P'(P(L)))&([S](L) △ {S € S|L C S} € P(S)). (1.1) If X is a nonempty set and X € P'(P (X)), then
(COV)[X|X] △ {x € P'(X)|X = J X} € P(P'(X))
xex
is the family of all coverings of the set X by sets of X; a covering k € (COV)[X|X] is called binary if k = {X1; X2}, where X1 € X and X2 € X.
Let R be the real line and N = {1; 2;...}; N € V'(R). For n € N, we use the notation T~n, = {k € N|k < n}. For a set H, denote by Hn the set of all mappings from 1,??, into H; thus, elements of Hn are ??.-tuples (hi)iey-^, where hj € H for j e 1,??..
Some special families. In this subsection, we fix a nonempty set I and consider families of P'(P(I)), i.e., nonempty families of subsets of I, especially, n-systems. More precisely, let
n[I] = {I€P'(P(I))| (0 €I)&(I €I)&(A n B €I VA €I VB €I)} (1.2)
be the family of all n-systems of subsets of I with "zero" and "unit." If I € n[I], then the pair (I, I) is considered as a measurable space in the wide sense. Denote by
n0[I] = {I€ n[I]| VL €I Vx € I \ L EIA €l: (x € A)&(A n L = 0)} the family of all separable n-systems of (1.2). Finally, we introduce the family
nj[I] = {I € n[I]| VE1 € I VS2 € I VS3 € I (1 3)
((E1 n X2 = 0)&(^2 n S3 = 0)&(£1 n £3 = 0^ (£1 n £2 n £3 = 0)}.
In (1.3), we have a special variant of n-systems connected with the linkedness property that plays a basic role in this paper. We introduce the family of lattices by using only subsets of I. More precisely, let
(LAT)0[I] = {L € n[I]| A U B € L VA € L VB € L}
be the family of all lattices of subsets of I. In addition, the following two particular cases are realized:
(alg)[I] = {A € n[I]| I \ A € A VA € A} = {A € (LAT)0[I]| I \ A €A VA € A},
(top)[I] = {t € n[I]| U G € t VG € p'(t)} = {t € (LAT)0[I]^ G € t VG € p'(t)}. GeG GeG
Certainly, (I, A) with A € (alg)[I] is a usual measurable space with algebra of sets. Moreover, (I,t) with t € (top) [I] is a topological space. Finally,
(clos)[I] = {F € (LAT)0[I]^ F €F VF' € P'(F)} = {Ci[t] : t € (top)[I]}. (1.4)
F eF'
Elements of (1.4) are closed topologies in the sense of P.S. Aleksandrov [18, Ch. 4, Sect. 1]. We introduce the family of all nonempty centered subfamilies of a n-system:
(Cen)[I] = {Z € P'(I)| p| Z = 0 VK € Fin(Z)} VI € n[I].
z ek
Now, we recall notions connected with bases and subbases of topological spaces. In addition,
(BAS)[I] = {p € P'(P(I)) | (I = U B) & (VB1 € p VB2 € p Vx € B1 n B2
Sep (1.5)
3B3 € P : (x € B3) & (B3 C B1 n B2))},
(cl - BAS)[I] = {P € P'(P(I)) | (I € P) & ( fl B = 0)
Sep (1.6)
& (VB1 € P VB2 € P Vx € I \ (B1 U B2) 3B3 € P : (B1 U B2 C B3) & (x / B3))}.
In (1.5), bases of sets open in some topology of I are introduced. Similarly, in (1.6), bases of sets closed in some topology of I are introduced. For brevity, we call bases from (1.5) and (1.6) open and closed, respectively. Certainely, the property {U}(^) € (top)[I] (the property {n}(^) € (clos)[I]) is realized for ft € (BAS)[I] (for ft € (cl — BAS)[I]). Moreover, we consider open and closed bases of a specific topological space. More precisely, for any topology t € (top)[I], let
((t — BAS)o[I] △ {5 € (BAS)[I] | t = {U}(^)}) &((cl — BAS)o[I;t] △ {£ € (cl — BAS)[I] | Ci[t] = {n}(^)})
(thus, open and closed bases of the topological space (I,t) are introduced). We introduce subbases of open and closed sets in some topology on the set I:
((p — BAS)[I] △ {x € P'(P(I)) | {n}„(x) € (BAS)[I]} = {x € P'(P(I)) | I = U X})
Xex
&((p — BAS)ci[I] △ {x € P'(P(I)) | {U}„(x) € (cl — BAS)[I]}).
Now, we introduce open and closed subbases of a specific topological space. More precisely, for t € (top) [I], let
((p — BAS)o[I;t] △ {x € (p — BAS)[I] | {n}„(x) € (t — BAS)o[I]}) (1 7)
&((p — BAS)0i[I;t] △ {x € (p — BAS)ci[I] | {U}„(x) € (cl — BAS)o[I;t]}).
In (1.7), the families of open and closed subbases of the topological space (I,t) are introduced. For t € (top)[I] and x € I, we set N°(x) △ {G € t|x € G} and
NT(x) △ {H € P(I)|3G € N°(x) : G C H}
obtaining the family of all neighborhoods of x in the topological space (I,t). In addition, for t € (top)[I] and A € P(I), we define the closure of A in the topological space (I,t):
cl(A, t) △ {x € I| A n H = 0 VH € Nr(x)}. In what follows, the supercompactness property plays an important role. In this connection, we set
((SC) — top)[I] △ {t € (top)[I]| 3S € (p — BAS)o[I;t] VG € (COV)[I|S] (1 8)
3Gi € G 3G2 € G : I = Gi U G2}.
Then, (1.8) is the family of all topologies converting I to a supercompact topological space. A supercompact T2-space is called [4] a supercompact set. Certainly, (1.8) can be defined in terms of closed subbases (see [4-9]).
2. The family of ultrafilters and its Stone equipping
In what follows, we fix a nonempty set E and consider its n-systems. Moreover, in this section, we also fix I € n[E]. Then
F*(I) △ {F € P'(I\{0})| (AnB € F VA € F VB € F)&(VF € F V/ € I (F C I) (I € F))}
is the set of all filters of the (widely understood) measurable space (E, I). The simplest variant of a filter is given by the following obvious definition: for x € E,
(I — triv) [x] △ {/ €I| x € /} € F*(I). (2.1)
In (2.1), we obtain the trivial filter corresponding to the point x. Maximal filters are called ultrafilters. Then
FQ(I) △ {U € F*(I) | VF € F*(I) (U C F) (U = F)} = {U € F*(I) | V/ € I (/n U = 0 VU € U) (/ € U)} (2.2)
= {U € (Cen)[I] | VV € (Cen)[I] (U C V) (U = V)}
is the set of all ultrafilters of the measurable space (E,I). It is known (see [19, (5.9)]) that
((I — triv)[x] € Fo(I) Vx € E) ^ (I € no[E]).
In the general case of I, VF € F*(I) EIU € FQ(I) : F C U (this property is established by the Zorn lemma). Thus, by (2.1), we obtain FQ(I) = 0. Let
(L) △ {U € Fq(i)| L € U} VL € I. (2.3)
In addition, (2.2) and (2.3) imply the following property:
(L) △ {U € FQ(I) | U n L = 0 VU € U} VL € I. (2.4)
Representation (2.4) plays an important role in the theory of ultrafilters. Note that
(UF)[E; I] △ {$!(/) : / € I} € npFftZ)].
In addition, (UF)[E;I] € n[Fo(I)] and, in particular, (UF)[E;I] € (BAS)[Fq(i)]. Thus, we obtain the topology of Stone type:
TI[E] △ {U}((UF)[E;I]) € (top)[FQ(I)].
Thus, (UF)[E;I] € (TI[E] — BAS)o[Fo(I)]. In addition, by (2.4), we have the following inclusion:
(UF)[E;I]) C TI[E] n Cf*(i)[TI[E]] (2.5)
(for I € (alg)[E], (2.5) becomes an equality, see [20, Remark 3.3]). Note that
(Fo(I), TI[E]) (2.6)
is a zero-dimensional T2-space (if I € (alg)[E], then (2.6) is a zero-dimensional compact set, see [20, Sect. 3]).
For H € P (E), we set
FC[i|H] △ {U € Fo(I)|3U € U : U C H}.
By (2.4), we have
FC[I|E \ /] = FQ(I) \ $I(/) V/ € I. (2.7)
Using (2.7), we obtain the following representation:
FC[I] △ {FC[I|A] : A € Ce[I]} = Cfs(i)[(UF)[E;I]] € (cl — BAS)o[Fo(I);TI[E]]. (2.8)
Thus, by (2.8), we obtain a closed base of the topological space (2.6). In this connection, it is useful to note one representation of ultrafilters connected with (2.7) and (2.8). More precisely, from (2.6)
and (2.8), we conclude that, for every L € I, the set (L) is the intersection of all sets of the family [Fo[I]]($x(L)). In addition, by [12, Proposition 3.1], for L € I and A € CE[I], we have
(L C A) ^ ($!(L) C FC[I|A]). As a corollary, we obtain the following property:
[FC[I]]($x(L)) = {FC[I|A] : A € [Ce[I]](L)} VL € I. (2.9)
From (2.9), we extract the useful equality
F0(I) = {U €F*(I)|V/€ I (I € U) V (3A€[Ce[!]](/): E \ A€ U)}. (2.10)
In connection with (2.10), we emphasize the constructions from [16, Sect. 2].
3. Ultrafilters and maximal linked systems; topologies of Wallman type
In what follows, unless otherwise stated, we fix a n-system E € n[E]. Since FC[E] € (p — BAS)[F0(E)], we introduce the topology of Wallman type
TE(E) = {U}({n}„(FC[E])) € (top)[F0(E)]. (3.1)
In addition, we obtain the topological space
(F0(E), T° (E)). (3.2)
Using (3.1), we also call (3.2) a topological space of Wallman type. To investigate the properties of this topological space, we consider maximal linked systems of the n-system E. More precisely, for X € P'(P(E)), we set
(X — link)[E] = {X € P'(X)| X1 n X2 = 0 VX1 € X VX2 € X}. (3.3)
Thus, (3.3) is the family of all linked subfamilies of X. Moreover, for Y € P '(P(E)), we consider the family of all maximal linked systems of the family Y:
(Y — link)0[E] = {Y € (Y — link)[E] | VZ € (Y — link)[E] (Y C Z) (Y = Z)}. (3.4)
By (3.4), the family
(E — link)0[E] = {S € (E — link) [E] | VS € (E — link)[E] (S C S) (S = S)} = {S € (E — link)[E]| VL € E (L n S = 0 VS € S) (L € S)}
is defined (see [12, Sect. 4]). Note that VS € (E — link)[E] € (E — link)0[E]: SC.S; see [12, (4.5)]. Moreover (see [12, (4.5)]),
VS € (E — link)0[E] VS € S V£ € E (S C £) (£ € S).
Note the following property [12, Sect. 4]:
F0(E) = {U € (E — link)0[E]| A n B €U VA € U VB € U} € P'((E — link)0[E]). (3.5)
For L € E, we set (E — link)0[E|L] = {S € (E — link)0[E]| L € S}. Moreover, for H € P(E), let
(E — link)Op[E|H] = {S € (E — link)0[E]| 3£ € S : £ C H}.
Now, we introduce the following families:
¿Op[E;E] △ {<E — link>op[E|A] : A € Ce[E]} € P'(P(<E — link)o[E])), (3.6)
¿Q[E;E] △ {<E — link)o[E|E] : E € E} = C^-Hn^E]^[E;E]].
In connection with (3.6), we note that ¿°p[E;E] € (p — BAS)[<E — link)o[E]]. As a corollary, the following topology of Wallman type is defined (see [12, Sect. 5]):
To<E|E> △ {U}({n}„( ¿op[E;E])) € ((SC) — top)[<E — link)o[E]].
In addition,
(<E — link)o[E], To<E|E)) (3.7)
is a supercompact Ti-space for which (see (3.5))
T° <E) = To<E|E)|f0 (£). (3.8)
Thus, (3.2) is a subspace of the topological space (3.7) and
¿Op[E; E] € (p — BAS)o [<E — link)o[E]; To<E|E)]. (3.9)
Now, we recall the useful corollary of the supercompactness of the topological space (3.7) connected with (3.9); more precisely, VG € P'(CE[E])
(<E — link)o[E] = U <E — link)op[E|G]) (3Gi € G 3G € G :
GeG
<E — link)o[E] = <E — link)op[E|Gi] U <E — link^E^]). More detailed information on the topological spaces (3.2) and (3.7) can be found in [12, 15, 16].
4. Ultrafilters and maximal linked systems; topologies of Stone type
We recall that, as a variant of (2.6), the zero-dimensional T2-space
(Fo(E), TE [E]) (4.1)
is defined. We consider (4.1) as a topological space of Stone type. Now, let us recall an analog of (4.1) for the space of maximal linked systems. To this end, note that
¿o[E; E] € (p — BAS) [<E — link)o[E]].
Therefore, the following topology is defined:
Tq<E|E) △ {U}({n}„ ( ¿Q[E; E])) € (top) [<E — link)o [E]]. (4.2)
In addition, topology (4.2) converts <E — link)o[E] to the zero-dimensional T2-space
(<E — link)o[E ], Tq <E|E)). (4.3)
Note that, by [12, Proposition 6.5], we have
TE [E] = Tq<E|E)|f0 (e ). (4.4)
Thus, in view of (4.4), we conclude that the topological space
(F0(E), TE [E ])
is a subspace of the topological space (4.3). Moreover, by [12, Proposition 7.1], we have the inclusion
T0(E|E) C T*(E|E).
If X is a nonempty set, t1 € (top)[X], t2 € (top)[X], and t1 C t2, then we call the triplet (X, t1, t2) a bitopological space. Then ((E — link)0[E],To(E|E),T*(E|E)) and (F£(E), TE(E), TE[E]) are two bitopological spaces. General properties of these two bitopological spaces are given in [12, 13, 16].
Now, let us recall some statements from [16, Sect. 7]. More precisely, to the end of this section, we suppose that E € n0[E]. Therefore (see Sect. 2), (E — triv)[x] € F0(E) Vx € E. As a result, we obtain the mapping
(E — triv)[.] = ((E — triv) [x]) xgE acting from E to F^(E) and the corresponding sets-images
(E — triv)[-]1(A) = {(E — triv)[x]: x € A} € P(F£(E))
for A € P(E). We can consider the closure of every such set both in the topology TE (E) and in the topology TE [E]. In addition (see [16, (7.2)]),
cl((E — triv)[-]1(A), TE[E]) = {U € F0(E)| A n U = 0 VU € U} VA € P(E).
Moreover, by [16, Proposition 7.1], we obtain
$E(£) = cl((E — triv)[-]1(£),TE(E)) = cl((E — triv)[-]1(£),TE[E]) V£ € E. (4.5)
From (4.5), we obtain the corresponding density property:
F0(E) =cl((E — triv)[-]1(E), TE (E)) = cl((E — triv)[-]1(E), TE [E]). (4.6)
Relations (4.5) and (4.6) imply the following fact: in the general case E € n0[E], the properties of the topologies TE(E) and TE[E] are similar. In [16, Sects. 5, 6], sufficient conditions for the coincidence of these topologies and conditions under which TE (E) = TE [E] are given as well as similar conditions for the topologies T0(E|E) and T*(E|E).
5. Proper maximal linked systems
Let us emphasize some properties of the set (E — link)0[E] \ Fq(E). We call elements of this set proper maximal linked systems. Let us observe some properties of such maximal linked systems (more details can be found in [15, 16]). We first note the following statement from [16]:
(E — link)0[E] \F0(E) = {S € (E — link)0[E]| 3S1 € S 3S2 € S 3Ss €S : S1 nS2 nS3 = 0}. (5.1) Hence, VM € (E — link)0 [E],
(A n B n C = 0 VA € M VB € M VC € M) (M € F0(E)). (5.2)
Moreover, (4.2) and (5.1) imply the following property:
(E — link)0[E] \ F0(E) € T*(E|E). Finally, using (5.1), we obtain the following equivalence:
((E — link)0[E] \ F0(E)= 0) ^^ (3£1 € E 3£2 €E ^£3 €E : (5 3)
(£1 n £2 = 0)&(£2 n £3 = 0)&(£1 n £3 = 0)&(£1 n £2 n £3 = 0)). ( . )
Theorem 1. The following equivalence holds:
(E € nQ[E]) ^ (<E — link)o[E] = Fo(E)). (5.4)
Proof. Let E € nQ [E]. Consider an arbitrary M € <E — link)o [E]. Let
(Mi € M)&(M2 € M)&(Ms € M). (5.5)
Then, by (1.3), we obtain the following implication:
((Mi n M2 = 0)&(M2 n M3 = 0)&(Mi n M3 = 0)) (Mi n M2 n M3 = 0). (5.6)
Now, we use (5.2). More precisely, by the choice of M, we have the following property (see (3.3)):
(Mi n M2 = 0)&(M2 n M3 = 0)&(Mi n M3 = 0).
From (5.6), we conclude that Mi n M2 n M3 = 0. Since the choice of (5.5) was arbitrary, we have the property
A n B n C = 0 VA € M VB € M VC € M. From (5.2), the inclusion M € Fo(E) follows. Thus,
<E — link)o[E] c FQ(E). (5.7)
Using (3.5) and (5.7), we obtain the equality <E — link)o[E] = Fo(E). Thus, we have the implication
(E € nQ[E]) (<E — link)o[E] = Fo(E)). (5.8)
Let <E — link)o[E] = FQ(E). Let us prove that E € nQ[E]. On the contrary, let E / nQ[E]. From (1.3), we obtain for some Ei € E, E2 € E, and E3 € E
(Ei n E2 = 0)&(E2 n E3 = 0)&(Ei n E3 = 0)&(Ei n E2 n E2 = 0).
Then, by (5.3), <E — link)o[E] \Fo(E) = 0, a contradiction. Thus, the required inclusion E € nQ[E] holds. Therefore, we have the implication
(<E — link)o[E] = FQ(E)) (E € nQ[E]).
Using (5.8), we obtain (5.4). □
Remark 1. Theorem 1 implies the following equality:
nQ[E] = {L € n[E]|<L — link)o[E] = FQ(L)}.
6. Supercompact spaces of ultrafilters
Consider some corollaries of Theorem 1. The n-systems from (1.3) play an important role in these constructions. The following statement is an obvious corollary of (3.8) and Theorem 1.
Proposition 1. If E € nQ[E], then T°<E) € ((SC) — top)[Fo(E)].
Proposition 1 motivates special investigation of specific variants of n-systems from 4[E], Note first the important property connected with the Cartesian product. Let us start with the simplest case. We need new notation. For nonempty families X and Y, we set
X{x}Y △ Wz) x pr2(z) : z eXxY}. (6.1)
Family (6.1) is nonempty. We can use n-systems of nonempty sets as X and Y• In addition, for all nonempty sets X and Y such that X € n[X] and Y € n[Y], we have
X{x}Y € n[X x Y]. (6.2)
Proposition 2. If X and Y are nonempty sets, X € n|[X] and Y € [Y], then
X{x}Y € n£[X x Y]. (6.3)
Proof. Fix X, Y, X, and Y satisfying our conditions. We have (6.2). Consider arbitrary sets
(Ai € X{x}Y)&(A2 € X{x}Y)&(As € X{x}Y)• (6.4)
Then, for some sets U1 € X, V1 € Y, U2 € X, V2 € Y, U3 € X, and V3 € Y, the following representation is valid:
(Ai = Ui x Vi)&(A2 = U2 x V2)&(A3 = U3 x V3). (6.5)
Let relations (6.4) satisfy the conditions
(Ai n A2 = 0)&(A2 n A3 = 0)&(Ai n A3 = 0). (6.6)
From (6.5) and (6.6), we obtain the following obvious properties: (Ui n U2 = 0)&(Vi n V2 = 0)&(U2 n U3 = 0)&(V2 n V3 = 0)&(Ui nU3 = 0)&(Vi n V3 = 0). (6.7) From (1.3) and (6.7), we obtain
(Ui n U2 n U3 = 0)&(Vi n V2 n V3 = 0). (6.8)
From (6.5) and (6.8), we obtain Ai n A2 n A3 = 0. Thus (see (6.6)), the following implication hols:
((Ai n A2 = 0)&(A2 n A3 = 0)&(Ai n A3 = 0^ (Ai n A2 n A3 = 0). Since the choice of (6.4) was arbitrary, we have proved (6.3) (see (1.3) and (6.2)). □
We now return to the fixed nonempty set E and suppose, unless otherwise stated, that E € nl[E] (see Proposition 1). In view of Proposition 1, we conclude that
(F0(E), TE(E)) (6.9)
is a supercompact Ti-space (see [12, Sect. 5]). Recall Remark 1: in our case,
(E- link)0[E]= F0(E). (6.10)
From (3.8) and (6.10), we obtain the following equality for the topologies:
TE (E) = To(E|E) (6.11)
(we recall that, in our constructions, E € n*[E]). Moreover, from (6.10), we obtain
(E - link)0p[E|H] = fC[e|H] VH € P(E).
As a corollary, we obtain the equality
¿Op[E; E]= FC [E ]. (6.12)
Proposition 3. If G € P'(Ce[E]), then
(Ff(E) =U fC[E|G]) (3G1 € G 3G2 € G : F0(L) = fC[E|G1 UFC[E|Gy). (6.13)
GeG
The corresponding proof follows from (6.10), (6.12), and the properties of the family ¿°p[E; E] mentioned in [12, Sect. 5]. In (6.13), we obtained a natural sharpening of Proposition 1.
Remark 2. Note that, by duality, the family (UF)[E; E] in the considered case is a closed binary subbase (see [12, (5.1)]) of the topological space (6.9). This property follows from [12, Proposition 5.1] and (6.12).
In the next section, we consider specific examples, for which (6.10) and (6.11) are realized. 7. Examples of supercompact ultrafilter spaces
We first consider the cases when E C R, where R is the real line (Proposition 2 enables extending our constructions to a vector case). Consider the case when this set E is bounded. More precisely, suppose that E is an interval of R. In addition, we will consider specific sets E.
Let us agree to use only square brackets in the notation of all intervals in R: ]x, y[, [x, y[, ]x, y], and [x, y] for x € R and y € R. In addition, relations between x and y can be arbitrary: x < y, x = y,
and y < x. Thus, for example, [1,0] = {{ € R|(1 — £)&(£ — 0)} = 0. In other words, we consider
the empty set as an interval. Moreover, for c € R, we set (as usually) ] — to, c[= {{ € R|£ < c} and
]c, TO[= {£€ R|c<£}.
For a € R and b € ]a, define
Ji(a,b) = {Jpr^z),p^(z)[: z € [a,b] x [a,b]},
J2(a,b) = {[pr^z), p^(z)] : z€ [a, b] x [a,b]},
J3(a,b) = {[pr1 (z),pr2(z)[: z € [a,b] x [a,b]},
J4(a,b) = {]pr1(z),pr2(z)] : z € [a,b] x [a,b]}. We obtain four nonempty families of subsets of R.
Proposition 4. If a € R and b € ]a, then
(J1(a,b) €nf []a, b[])&(J2(a, b) € nf [[a,b]])&(J3(a,b) € nf [[a,b[])&(J4(a,b) € nf []a,b]]). (7.1)
P r o o f. The proofs of all four statements in (7.1) are similar. Therefore, let us consider only one of them and establish the property J1(a, b) € nf []a, b[], where a € R, b € R, and a < b. Note the obvious property:
J1(a,b) € n[]a, b[]. (7.2)
Let A1 € J1(a, b), A2 € J1(a, b), and A3 € J1(a, b). Then, for some numbers u1 € [a, b], u2 € [a, b], v1 € [a, b], v2 € [a, b], w1 € [a, b], and w2 € [a, b], we have
(A1 =]U1,U2D&A2 =]V1,V2p&A3 =]W1,W2Q. (7.3)
Let (A1 n A2 = 0)&(A2 n A3 = 0)&(A1 n A3 = 0). We introduce ^ = sup({u1; v1; w1}) € [a, b] and v = inf({u2; v2; w2}) € [a, b]. Then, by (7.3), v[c A1 n A2 n A3 and ^ < v (the latter inequality
follows from the linkedness of the family (Ai; A2; A3}). As a result, Ai n A2 n A3 = 0. Thus, we obtain the implication
((Ai n A2 = 0)&(A2 n A3 = 0)&(Ai n A3 = 0)) (Ai n A2 n A3 = 0). (7.4)
Since Ai, A2, and A3 were chosen arbitrary, the required property of the family Ji(a, b) follows from (1.3), (7.2), and (7.4). □
Thus, by Propositions 2 and 4, we obtain many examples of realization of (6.10) and (6.11). In these examples, (6.9) is a supercompact Ti-space (see Proposition 1). Later, we will provide analogs of (7.1) for the case E = R. Now we first observe a very simple general construction. Unless otherwise stated, we suppose that E is an arbitrary nonempty set. Suppose that
no △ {L€P'(P(E))| (0 €L)&(A n B € L VA € L VB € L)}. (7.5)
Certainly, n[E] = {L € |E € L}. It is obvious that L U {E} € n[E] VL € . Moreover, let
nE △ {L € no| VLi € L VL2 € L VL3 € L (7 6)
((Li n L2 = 0)&(L2 n L3 = 0)&(Li n L3 = 0)) (Li n L2 n L3 = 0)}.
Then, we obtain the following obvious property:
L U {E} € nQ[E] VL€ nE. (7.7)
In (7.5)-(7.7), we have simple modifications of definitions of Section 1 (see (1.2) and (1.3)). Consider now the case when E = R. In addition,
(jRi) △ {]pri(z), p^(z)[: z € R x R} € nR) &(JR2) △ {[pri(z), pr2(z)]: z € R x R} € nR) (7 8)
&(JR3) △ {[pri(z), pr2(z)[: z € R x R} € nR) &(JR4) △ {]pri(z), pr2(z)] : z € R x R} € nR).
The corresponding proof is an obvious analog of constructions used in the proof of Proposition 4. By (7.8), we obtain new variants of (6.10), (6.11), and Proposition 1 for the case when E = R and
(E = jRi) U {R}) V (E = jR2) U {R}) V (E = jR3) U {R}) V (E = jR4) U {R}).
The direct combination of Propositions 2 and 4 and similar combination of Proposition 2 and (7.8) enable obtaining variants of (6.10), (6.11), and Proposition 1 for subsets of R x R.
8. The case of generalized Cartesian product
In this section, we consider a natural generalization of Proposition 2. More precisely, we fix nonempty sets X and E. In what follows, for every nonempty set S, we denote by (as in [17, Ch. II, Sect. 6]) the set of all mappings from X to S; we can use a nonempty family as S. In addition, we fix a (set-valued) mapping (Ex)xeX € P'(E)X; thus, we have a mapping
x ——> Ex : X —>P'(E). (8.1)
Using (8.1) (and the axiom of choice), we obtain the (nonempty) set
E = J] Ex = {/ € EX|/ (x) € Ex Vx € X} € P'(EX). (8.2)
xex
In connection with (8.1) and (8.2), note that
n[Ex] = {(Lx)xex € P'(P(E))x|Ly € n[Ey] Vy€X}. (8.3)
xex
Similarly, we define the set
[] nf[Ex] = {(Lx)xex € P'(P(E))X|Ly € nf[Ey] Vy € X} € P([] n[Ex]).
xex xex
From (8.3), we obtain the following obvious representation:
H Lx = {(Lx)xex € P(E)X| Ly € Ly Vy € X} € P'(P(E)X) V(Lx)xex € [] n[Ex]. (8.4)
xex xex
As a corollary, by (8.4), the following product of sets is defined:
n Lx = {/ € EX|/(x) € Lx Vx € X} V(Lx)xex € J] n[Ex] V(Lx)xex € J] Lx. (8.5)
xex xex xex
Now, we introduce two variants of the n-system product: usual (similar to the variant applied
in the general topology in the Tychonoff product construction) and "box" variants. The latter
variant corresponds to the way used in the construction of the box topology.
More precisely (see (8.4) and (8.5)), we introduce the usual n-system product
for (Lx)xex € n n[Ex]: xex
0 Lx = {H € P(EX)|3(Lx)xex € n Lx : (H = n Lx)&(3K € Fin(X) :
xex xex xex
Ls = Es Vs € X \ K)} = {H € P(E)|3(Lx)xex € n Lx : (8.6)
xex
(H = 11 Lx)&(3K € Fin(X): Ls = Es Vs € X \ K)}.
xex
The "box" variant is defined easier. For (Lx)xgX € n n[Ex], we set
xex
O Lx = { n Lx : (Lx)xex € J] Lx }. (8.7)
xex xex xex
We note two obvious (but useful) properties. More precisely, for (Hx^)xeX € P(E)X and (Hx2) )xex € P(E)X, we have
( n Hx1)) n ( n Hx2)) = n (Hx1) n Hx2)). (8.8)
xex xex xex
Similarly, for (H^)x€x € P(E)X, (^2))x€x € P(E)X, and (Hx3) )x€x € P(E)X, we have
( n Hx1)) n ( n Hx2)) n ( n Hx3)) = n (Hx1) n Hx2) n Hx3)). (8.9)
xex xex xex xex
From (8.6) and (8.8), we obtain
0 Lx € n[E] V(Lx)xex € [] n[Ex].
xex xex
Moreover, (1.2), (8.7), and (8.8) imply
O Lx € n[E] V(Lx)xex € [] n[Ex]. (8.10)
xex xex
In addition, we obtain the following obvious inclusion for (Lx)xeX € n n[Ex]:
xex
0Lx ^QLx. (8.11)
xex xex
Theorem 2. If (Lx)xex € n nl[Ex], then
xex
(0 Lx € 4[E])&(Q Lx € n»[E]). (8.12)
xex xex
Proof. We restrict ourselves to the verification of the latter inclusion in (8.12) (the corresponding verification of the former inclusion is an obvious corollary; see (8.11)). In this proof, we fix
(Lx)xex € II n|[Ex] (8.13)
xex
and use (8.10). From (8.10) and (8.13), we obtain
(Lx)xex € n[Ex] : L = © Lx € n[E]. (8.14)
xex xex
Thus, by (8.7) and (8.14), we have the following equality:
L = { Lx : (Lx)xex € L4. (8.15)
Jx ■ V-^x/xfcX ^ I I >— x i
xex xex
Consider arbitrary sets Ai € L, A2 € L, and A3 € L. Taking into account (8.15), we choose
€ Lx)&(Vx) xex € Yl Lx)&(Wx) xex € Lx) xex xex xex
with the following properties:
(Ai = [] Ux)&(A2 = [] Vx)&(A3 = [] Wx). (8.16)
xex xex xex
Then, by (8.8) and (8.16), we conclude that Ai n A2, A2 n A3, and Ai n A3 are realized as follows:
(Ai n A2 = JJ(Ux n Vx))&(A2 n A3 = JJ(Vx n Wx))&(Ai n A3 = JJ(Ux n Wx)). (8.17)
xex xex xex
Similarly, from (8.9) and (8.16), we obtain
Ai n A2 n A3 = H (Ux n Vx n Wx). (8.18)
xex
Suppose now that
(Ai n A2 = 0)&A2 n A3 = 0)&(Ai n A3 = 0). (8.19)
Then, from (8.17) and (8.19), we obtain, for x € X,
(Ux n Vx = 0)&(Vx n Wx = 0)&(Ux n Wx = 0), (8.20)
where Ux € Lx, Vx € Lx, and Wx € Lx. However, for x € X, the inclusion Lx € n|[Ex] holds; therefore, by (1.3) and (8.20), we have Ux n Vx n Wx = 0. Using (8.18) (and the axiom of choice), we obtain A1 n A2 n A3 = 0. Thus, we have proved the implication
((Ai n A2 = 0)&(A2 n A3 = 0)&(Ai n A3 = 0)) (Ai n A2 n A3 = 0). (8.21)
Since Ai, A2, and A3 were chosen arbitrary, by (8.21), we obtain (see ( )) L € 4[E]. By ( ), we have
O Lx € n»[E].
xex
Thus, the theorem is proved (see (1.3) and (8.11)). □
Now, we observe some cases connected with the realization of our measurable space equipped with the semi-algebra of sets. These measurable spaces are widely used in the probability theory (see [1]). Let us first give a general definition in terms of n-systems. For this, we fix a nonempty set X. If L € n[X], A € P(X), and n € N, then
n
An(A,£) = {{Lt)l&T7^Cn\ (A= \jLi)k(LpnLq = 0 Vp &TJi Vç<E M \ {?})}.
i=i
We set
n[X] △ {L€ n[X]| VL € L 3n € N : An(X\ L, L) = 0};
thus, the family of all semi-algebras of subsets of X is introduced. Certainly, (alg)[X] c n[X]. Moreover,
nf [X] △ {L € n[X]| VLi € L VL2 € L VL3 € L ((Li n L2 = 0)&(L2 n L3 = 0)&(Li n L3 = 0)) (Li n L2 n L3 = 0)} (8.22)
= n[X] n nf[X].
Elements of family (8.22) are special semi-algebras of subsets of X. Note that, for a € R and b €]a, by Proposition 4, J3(a, b) € nï [[a, b[] and
([a, b[, J3 (a, b)) (8.23)
is the usual space-pointer. Of course, (8.23) is a specific measurable space with a semi-algebra of sets. In addition, by Proposition 4, J3(a, b) € nf[[a, b[]. Thus, for E = [a, b[ and E = J3(a, b), we obtain (6.10) and (6.11).
Consider a more complicated case. We return to (8.1) and (8.2). For a nonempty set X and (Ex)xeX € P'(E)X, we introduce
H n[Ex] △ {(Lx)xex € P'(P(E))X|Ly € n[Ey] Vy € X} € P( [] n[Ex]). (8.24)
xex xex
By (8.14) and (8.24), we define
0 Lx € n[E] VLx)xex G H n[Eœj.
Moreover, the following property holds:
0 Lx G n[E] VLx)xex G [] n[E].
(8.25)
From Theorem 2 and (8.25), we obtain for (Lx)xgX € n nl[Ex]
xex
0Lx G n|[E].
(8.26)
xex
Note an obvious corollary concerning the case E = R.
Proposition 5. Let (ax)xeX € RX and (bx)xgX € n ]ax, Then
xex
0 J3(ax,bx) G [{/ G RX| Vx G X (ax < / (x))&(/ (x) < bx)} .
xex
The corresponding proof is obtained by the direct combination of (8.26) and the above-mentioned properties of the measurable space (8.23). Thus, by (8.22) and (8.25), we obtain one more variant of realization of (6.10) and (6.11) in the class of measurable spaces with a semi-algebra of sets (see Proposition 1).
In this paper, relations between ultrafilters and maximal linked systems on a n-system were considered. The emphasis was on the conditions under which all maximal linked systems are ultrafilters on the n-system. The investigation of the ultrafilter space is important, since in the class of ultrafilters the density properties (see (4.5) and (4.6)) are realized. Therefore, this space can be considered as a kind of extension of the initial set E whose elements play the role of usual solutions. Such interpretation is very natural for extension procedures of extremal problems and attainability problems (see [2, 3, 19, 20]). More precisely, the ultrafilters can be used as generalized elements. Therefore, it is useful to know as much as possible properties of topological nature. The supercompactness is a very interesting property of such type. We investigate this property identifying ultrafilters and maximal linked systems for a n-system. Of course, this approach is not universal. However, we obtain a positive result for many interesting variants of n-systems. In addition, we use the supercompactness property of the space of maximal linked systems with Wallman equipping.
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9. Conclusion
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