On modular filtersAbstract. Modular filters and their special cases are investigated. Some properties of families

Séria I: PRACE M A T E M A T Y C ZN E XX IX (1989)

Ed w a r d Am b r o z k o (Poznan)

On modular filters

Abstract. M odular filters and their special cases are investigated. Some properties of families of modular filters are proved. Theorems on generating modular filters are given. Lattices of modular pseudotopologies are discussed.

We continue the investigations of [1]. Our terminology and notation appearing in this paper are kept.

Let X be a linear space over the field К of real or complex numbers.

1. Modular filters. Denote by LP(X) the set of all linear pseudotopologies ([3], [4], [5]) defined on X. In [1] we considered linear pseudotopologies

t eLP(X) satisfying the following condition:

(M) There exists a filter 3F gt(0) such that for every ^ e t(0) the inclusion X^ c= ^ holds with a certain number XeK.

Another form of condition (M):

There exists a filter ^ e F ( X ) (the set of all filters in X) such that

t(0) = {&eF(X): a# - c= ^ with a certain XeK}.

In condition (M) we may assume that X ф 0 ([1], 5.1.2).

1.1. Let be filter-bases in X. We define the following relations (cf. with [7], [9]):

@ x if and only if c= a[J^2] with а certain number а ф

0

;

~ 2 if and only if -<

^ 2

and

Recall that for a filter-base 0!t the symbol [J 1] denotes the filter generated in the given set by J*. The relation ■< is reflexive and transitive, ~ is an equivalence relation (on the set B(X) of all filter-bases defined in X).

1.2. Let a pseudotopology t gLP(X) satisfy condition (M) with a filter & . Then this pseudotopology satisfies (M) also with a filter У e F(X) if and only if

& ~ У .

Proof. Suppose that the pseudotopology т satisfies condition (M) with a filter . Of course, ^ and 3F =э with certain numbers a, f Ф 0.

Therefore, 3F ~ <3. Now, let &eF(X) and let ^ . Then a 3F c for

some a, fit К, a, f Ф 0. From the inclusion Ж c= we infer that f ет (

0

).

Take a filter J'Fei(O). Then Ж => Я#-, where Я is a certain number. Thus Ж => аЯ^.

1.3. //а /шеаг pseudotopology a satisfies condition (M) with a filter Ж , then we write о = x^ (the meaning of the symbol x^ is different from the meaning of the symbol x$- introduced in [

1

]).

Recall two theorems of [1]:

Let (X , t) be a balanced linear-pseudotopological space satisfying (M) with Ж . Then the following conditions hold:

(bMl) there exists a number p Ф 0 such that lЖ + 1Ж о рЖ\

(bM2) Vx з Ж for every x e X .

Suppose that ЖеР(Х). Moreover, let Ж satisfy conditions (bMl) and (bM2). Then the equalities

t(0):= (^ eF (X ): ^ => ЛЖ for a certain ЛеК, Я ф 0}, т(х) := т(0) + х, x e X ,

define a balanced linear pseudotopology on X satisfying condition (M).

We now introduce some special cases of filters satisfying (bMl) and (bM2).

First admit for a set Я in I the symbols ([7], [9]):

A(A):= IA + IA,

r s{A)\= {ax + py: x , y e A ; а, jSeK; |a|s + |£|s ^ 1}, where s > 0 (Г(А): = rf A) ) .

If s > 0 and Же F (X), then the families {A(F): Ре Ж} , (FS(F): Р е Ж } are filter-bases in X; therefore, we define:

А{Ж): = [ {A(F): Р е Ж } \

r , ( ^ ) : = [ { r s(F): FeJF}] (Г(&)'■ = Г ,(#')).

It is obvious that А (Ж) = IЖ + 1Ж .

1.3.1. Let a number у > 0 be given. We say that Ж eF(X) is a (generalized) modular filter of the character у in the space X if it satisfies condition (bMl) with у and condition (bM2).

A base of a modular filter of a character у > 0 will be called a modular base of the character y.

1.3.2. Observe that a filter Ж eF(X) satisfying (bM2) and (bMl) with p ф 0 satisfies condition (bMl) with every ЛеК, where |Я| = \p\; hence, this filter is modular of the character \p\.

1.3.3. Let Ж eF(X) be a modular filter of a character у > 0. Then the filter IЖ is also modular of the character y; moreover, IЖ ~ Ж .

Proof. We have: A (IF) = A{,F) ^=d y F ^=> y l F^; Vx ^{i d} F ^{= э} for x e l ;

id /JF-id J (,jF) id y#" (hence ~ J*7).

1.3.4. Let 0 < yj < y2. Then every modular filter of the character yx is also of the character y2.

Proof. Suppose that F^F(.X) is a modular filter of the character yx.

Since, moreover, 0 < y j y 2< 1, we have (yJyf)A(F) ^{i d} A(F) ^{i d} yxF '. There­

fore, A(F) => y2F , and consequently our filter is modular of the character y2.

1.3.5. Let s > 0 be given. We say that a filter F eF(X) is s-modular in the space X if it satisfies (bM2) and the following condition: & cz TS(F) (i.e., F = TS(F), because the inclusion ГS(F) cz F is always satisfied).

A base of an s-modular filter (s > 0) is called an s-modular base (cf. with [9]).

Observe that an s-modular filter (s > 0) is balanced.

1.3.6. Let 0 < sx < s2. Then every s2-modular filter is also sx-modular.

Proof. Suppose that a filter F e F ( X ) is s

2

-modular. Since for any set А с X the inclusion TSi(A) cz TS2(A) holds, we have F cz TS2( F ) <= TSi(F).

1.3.7. Every s-modular filter (s > 0) is modular of the character 21/s.

Proof. Let F e F ( X ) be an s-modular filter, where s > 0. For all A cz X the inclusion 2_

1

/s(L4 + IA) cz r s(A) holds (because if |a|, \f}\ ^ 2~1/s, then

|a|s + |/?|s < 2

-1

+ 2

_1

= 1), so r s(F) <=: 2~1/SA(F). Moreover, F cz Г S(F), therefore F cz 2~1/sA(F), i.e., 21/sF cz A(F).

1.3.8. Let s > 0, U с X be given. We say that the set U is absolutely s-convex if r s(U) cz U. If a set is absolutely 1-convex, then we say that it is absolutely convex.

Observe that every non-empty and absolutely s-convex set, where s > 1, is a subspace of the given linear space.

If 0 < sx < s2, then an absolutely s2-convex set is also absolutely Sj -convex.

Of course, an absolutely s-convex set (s > 0) is balanced.

For a set U cz X and for a number s > 0, by absconvst/ we denote the smallest absolutely s-convex set containing U; we have

abscon vsU

= {cc1x1+ ... +anx„: neN; al5 ..., a„gK; laj'-b ... +|a„|s ^ 1; xx, ..., xneU}.

Obviously, N is the set of all positive integers.

1.3.9. We say that & e B(X) is a locally s-convex modular base (s > 0) in the space X if every set of F is absolutely s-convex and absorbent in X (cf. with [7]).

1.3.10. If &eB(X) is a locally s-convex modular base (s > 0), then [J*]

(defined in X) is called a locally s-convex modular filter.

1.3.11. Every locally s-convex modular filter (s > 0) is s-modular.

1.3.12. If 0 < Si < s2, then a locally s2-convex modular filter is locally s^convex.

1.3.13. The neighbourhood filter of 0 in a linear-topological space is called an LT-filter; any its base is called an LT-base ([7]).

Observe that a linear topology satisfies condition (M) only with its LT-filter.

1.3.14. (1) If F eF(X) is a modular filter of a character ye(0, 2), then F is an LT-filter.

(2) If F e F\X) is an s-modular filter, where s > l, then F is an LT-filter.

(3) If F eF(X) is an LT-filter, then F is a modular filter of the character y for any y >

0

and an s-modular filter for any s >

0

.

Proof. (1) Let F eF(X) be a modular filter of a character ye(0, 2). We have 2 F zd I F + I F zz> y F , so y F c= 2F . Since 0 < y < 2, F is an LT-filter ([1], 5.4.3).

(2) An s-modular filter, where s > 1, is modular of the character 21/s e (l, 2), therefore it is an LT-filter.

(3) Let F be an LT-filter in X. Then for any y > 0 we have I F + I F .= F + F = F = y F , so the filter F is modular of the character y. Now take a number s > 0. Since {(o t , f ) e K x K : \a\s+ \f\s ^ \} cz I x I , for every set F <zz X we have TS(F) <= A(F). Thus r s(F) zz> A(F). Moreover, A(F) => F , therefore FS( F ) гэ F .

1.3.15. Let numbers y, s > 0 be given and let the symbols FMy{X), FSM(X), FMs_cXX), FLT{X) denote the set of all modular filters of the character y in X , the set of all s-modular filters in X, the set of all locally s-convex modular filters in X and the set of all LT-filters in X, respectively. By virtue of our lemmas we obtain the inclusions: FSM(X) czFM2i/s(X), FMS.C (X) c F M(X), FLT{X) c FMJX) n FSM(X), FM (X) C= FM72(X) for 0 < Yl < y2, FS2M( X) C= FS1M(X) and FMS2.JX) c= FMSl.c_(X) for 0 < Sj < s2. Moreover, let us note that: TMS.C,(X) = {{X }} for s > 1, FMy{X) = FLT{X) for ye(0, 2), FSM(X) = FLT(X) for s > 1.

1.3.16. Let numbers y, s > 0 be given. We say that a pseudotopology

teLP{X) (and the space (X, t)) is modular of the character y, s-modular, modular and locally s-convex if there exists a filter F belonging to FMy{X), FSM(X), FMS_C {X), respectively, such that т = x^.

1.3.17. We have proved that a linear pseudotopology t eLP(X) satisfying (M) with a filter gF(X) satisfies the one also with a filter ^ e F (X) if and only if 3F ~ 3 (1.2). Now give some properties of the relation

(1) If a filter 3? eF(X) is modular of a character y ^ 2, 3 e F( X), 3 ~ 3F f/ien the filter 3 is modular of a certain character; however, у need not be a character of 3 .

Proof. Suppose that a filter J^gF(X) is modular of a character у > 2 and

^ e F (J ), 3 ~ Then =э Fc => J5- for every x e X and vf3 c for certain numbers a, ft ф 0. Hence

d(^) =э ^A (^) =з =з

This means that d(^) з |ay//?|0. Moreover, Fc => 3F => for any x e X ; therefore Fc =э ^ for x e X .

Let us give an example of equivalent modular filters having different characters. We take X : = R over R (the space of real numbers), fF : = [< — 1, 1 )], 3\ — [< — 1, 2 )]. The filter is modular of the character 2, the filter 3 is modular of the character 4 and this character cannot be decreased (13 + 13 = [< — 4, 4 )] =з 4[< — 1, 2)]). The filters 3F, 3 are equivalent (

2

[< —

1

,

1

>] <= [< —

1

»

2

>] cz [< —

1

,

1

>]).

(2) If 3F eFLT(X) and 3 eF(X), 3 ~ , then, obviously, 3 = 2F.

(3) If ^ e FMS_C (X), s > 1 and 3e F( X) , 3 ~ 3F, then, of course, 3

= * = {X}.

(4) The following situations are possible:

P e F ^ M i X ) , s0s(0, 1>, 9eF(X), <g ~ , <S$ (J F.M(X);

F e F M ^ X X ) , s0 e (0, 1>, <SeF(X), <S ~ ? , <Si{} FM„„(X).

s > 0

As an example for (4) we may take the one of (1).

Let us give an example of applications of the above remarks: If a linear pseudotopology т is modular of a character У>

2

, г = (and if it is no linear topology), then the filter 3 is modular of a certain character, but у need not be a character of 3 .

1.4. In [

8

] and [9] modular bases equivalent to countable (in particular finite) ones are investigated. These bases can be (in an appropriate manner) described by pseudomodulars.

We give an example to show that a filter equivalent to one having a countable base can have no countable base.

X : = R over R,

# " := [{< —2, 2 У\А: A c=. < — 2, 2), A is countable, \a\ > 1 for a g A}], 3\ = [< — 1, 1>].

The filter ^ is modular of the character 2 and, obviously, it has a countable base. Moreover, ~ <§ (3F <= ^ cz^#-). The filter has no countable base. It is modular of the character 4 (/# ' + /# ' э 4 # ) .

2. Filters satisfying condition (bMl). Give some preliminary remarks.

2.1. If a pseudotopology on a linear space is linear, then we say that this pseudotopology is consistent with the algebraic operations of the given space (shortly: is consistent with the space; is consistent). Let us give a more general notion.

2.1.1. We say that a pseudotopology т on X is weakly consistent (with the algebraic operations of the space X; with the space X) if the following conditions are satisfied ([5]; X is equipped with the pseudotopology t):

the addition + : X x X -> X is continuous,

the multiplication •: K x X ^ - X is continuous at (

0

,

0

),

for every number AeK the mapping f k: X - + X such that f k(x): — Ax, x e X , is continuous.

2.1.2. A weakly consistent pseudotopology т on X satisfies conditions (a), ((3), (1), (2), (3), t(x) = t(0) + x (x eX) adduced in [1], 2.1 (cf. with [5], 5.7.7).

2.1.3. Let a non-empty family t(0) c: F(X) satisfy conditions (a), (P), (1), (2), (3) of [1]. Moreover, let т(х):= t(0) + x for x eX. Then the mapping т is a weakly consistent pseudotopology on X.

2.1.4. We say that a weakly consistent pseudotopology т is balanced (equable) if for every filter ^ет(О ) there exists a filter ^ет(О) such that

& ZD У = I<S (& dF^).

Observe that a weakly consistent pseudotopology т is balanced if and only if I ^et(0) for every ^ет(О).

2.1.5. Let a non-empty family т(0) c F(X) satisfy conditions (a), (p), (1), (2), (3b) of [1]. Moreover, let т(х):= t(0) + x for x e X . Then the mapping т is a balanced weakly consistent pseudotopology on X (cf. with 3.6 of [1]).

2.1.6. Suppose that a pseudotopology т on X is weakly consistent. Then (i) the set Y: = { x e X : Кхет(0)} is a linear subspace of the space X.

Moreover, let

i: Y-+X, i(x):= x for xeY,

<r(0):= { ^ 'eF(Y)\ *(^)ет(0)}, <r(x):= <r(0) + x for x eY.

Then

(ii) a is a linear pseudotopology on Y,

(iii) the mapping i: (Y, o ) ^ ( X, z) is continuous,

(iv) every pseudotopology g on Y,for which the mapping i: (У, @)->(X, t) is continuous, satisfies the condition д ^ о .

Proof. Let us take a certain filter ^ £

1

(

0

). We have OJ5- = [0] = F0 ei(0); so 0e Y, and therefore Y ф 0 . Let x, y e Y and XeK be given. Of course, [Fx]x, [Fy]x £i(0) (the symbol [Fx]* means that the filter is constructed in X).

If e > 0, then clx + sly => e/(x + y); therefore [Fx]y + [Fy]y <= [F(x + y)]x. Hence we get [F(x + y)]x £t(0), and thus x + ye Y. Moreover, [F(2x)]x = A[Fx]x£t(0), so XxeY.

We now show that a is a linear pseudotopology on Y. Consider the appropriate conditions ([3], [4], [5], [1], 2.1).

(a) Suppose that F(Y)b3F =э ^ео(0). Then F(X)3i(^) (= [ ^ ] x) i(@)e t(0); so i(^)ex(0). This means that 3F£<j(0).

(P), (1) Let filters 3F, &ecr(0) be given. Then we have i(3F), i(^)ez(0);

therefore i(^F) n î(^ )£t(0). Moreover, i{3F nf?) = i(3F) n if#), so 3F n &eo(0).

On the other hand, we get i(,3F + <&) = i(^ ) + i(('S) £ t(0). Therefore 3F + ^ £ cr(0).

(2), (3) Let a filter ^ £er(0) be given. Obviously, i(^)ex(0). Take a number XeK. Since i(X^) = Xi(#")et(0), we have X ^eg(0). Moreover, i(FW)

= Vi(^F)e t(0); therefore VWeg(0).

(4) Let an element x £ Y be given. By virtue of the condition i([Fx]y)

= [Fx] * £t(0) we infer that [Fx]y£cr(0).

Let x eY, 3F £<t(x) be given. Then i ( —x)£i(0). So we have i{3F) = i{3F — x) + x£t(x). Therefore the mapping i: (Y, a ) ^ ( X , x) is con­

tinuous.

Take a pseudotopology g on Y such that the mapping i: ( У, g) ->(X, t) is continuous. Moreover, let xeY, ^ ед(х)Ъе given. Then i{3F)et(x); therefore 3Feо (x) (we have i{3F — x) = i(^ ) — x et(0), so3F — x£cr(0)). Thus g(x) c= er(x) for x £ У, i.e., a ^ g.

Observe properties (iii), (iv) mean that (У, a) is a subspace of the space (X, r). The pseudotopology x was weakly consistent, but the pseudotopology a is consistent.

2.2. Give the following theorems on weakly consistent pseudotopologies (cf. with [1], 5.3.2, 5.3.3):

2.2.1. Let a weakly consistent pseudotopology x on X be balanced and let 3FeF(X) be a filter such that

t(0) = {@eF(X): У => XXsF for a certain XeK}.

Then the filter 3F satisfies condition (bMl).

The proof is a part of the one of the similar theorem given in [1]. Observe only that we may assume that X Ф 0. In fact, suppose that (X^ a ^ e F(X), i.e., & = [0]x . Since 13? e i(0), the inclusion <= I3F holds for a certain

number a. Therefore о /#" c= 2F. So: if a = 0, then ,F = [0]; if a # 0, then

F с - I F cz [

0

].

a

Thus we have always c= [0], and hence ^ = [0] => J* = 1#\

2.2.2. If a filter F eF(X) satisfies condition (bMl) and

t(x): = {&eF(X): У => Я#- with a certain number Я ^ 0} + x for x eX, then

t is a balanced weakly consistent pseudotopology on X.

2.3. Obviously, for the pseudotopology defined on X as in 2.2.2 we can construct the subspace Y and the pseudotopology a in the manner given in

2

^.

1

^.

6

^.

2.3.1. Let a filter F eF(X) satisfy condition (bMl) with a number ц ф 0.

Then this filter satisfies (bMl) with every v e K , where |v| = |/r|. We say that F is a modular filter of the character \ц\ in X (a modular filter without property (bM2)). Let us construct the pseudotopology x of 2.2.2 for our filter F and the subspace (Y, cr) of 2.1.6. We have:

t(x):= eF(X): У =э ЯF with a certain Я # 0 } + х, x eX;

Y:= { x e X : Кхет(0)} = (x e V : Vx => I F with a certain Я ф 0}

= { x e l : К хз

= ( x e l : for every F e# " there is a number а Ф

0

such that a x e F}

(cf. with [1], 5.3.4). Observe that if 08 is a base of the filter F , then Y = ( x e l : for each Be & there is an a # 0 such that olxeB}.

Consider the family

F\Y: = { F n Y : FeF ) .

We maintain that (F\ Y)eFM\/t\(Y). First show that F\Y is a filter. Since 0eF n Y for every FeF , 0^{F\ Y). Let sets A, A'e(F\Y) be given. Then we have A = F n Y and A' = F ' n Y for certain F, F'eF . Moreover, A n A'

= ( FnF' ) nY, F n F'eF ; therefore A n A'e(F\ Y). Now suppose that Ae(F\Y) and A с B c Y. Of course, A = F n Y with some FeF . By virtue of the conditions В = (В и F) nY, BkjFeF we infer that Be(F\ Y). We have proved that (F\ Y)eF(Y). Verify condition (bMl) for the filter F\Y. Take a set Ae{F\ Y). It has the form A = F n Y, where F eF . Obviously, therç exists a set F' e F such that -A(F') c= F. Therefore -A(F' n Y) cz F n Y (recall that

В В

Y is a linear space). Thus the filter F\Y satisfies (bMl) with ц (also with |//j).

Now take a set Ae(F\Y) and an element xeY. Moreover, let A = Fn Y, where F e F . Since х е У and F e F , there exists a number а ф 0 for which

olxgF. Furthermore, axe Y (Y is a linear space); therefore axe A, and ([1], 5.3.4) the filter F\ Y satisfies condition (bM2).

Next show that

cr(0) = (^eF(Y ): ^ => X{F\Y) for a certain XeK, X # 0}, i.e., (Y, a) is a modular space of the character \fi\. In fact, we have

^eer(O) if and only if ^eF (Y ) and i(^) n> XF

for some number X ф 0.

Moreover, we will show that for (SeF{Y) and XeK the equivalence

i f i ? ) zdXF if and only if ^ zdX(^\Y)

holds. Let &eF(Y) and XeK be given. Suppose that ifi0) => XF and take a set AeX(F\Y). Then there is an F e F for which X( FnY) czA. Moreover, G cz XF for some G e f . Thus X(F n Y) = (XF) n Y z z G n Y = G , and we have Y^zdA ^zd G e^ , i.e., А е У . We have obtained the inclusion X(F\ Y) . Now suppose that X(F\ Y) <= ^ and take a set A e X F . Then A ^zdXF for a certain Fe F . Furthermore, there is a set G e^ such that X{F n Y) з G, so we have X ^zdA ^zdXF ^zdX(F n Y ) zz> Ge&, i.e., Aeifi?). Therefore XF c= i(Xg).

2.3.2. If a filter F eF(X) satisfies the condition F cz r s(F) for some s > 0, then it satisfies (bMl) with ц = 21/s and (F\Y)e FSM(Y).

Proof. Suppose that F eF(X), XF c= FfXF), s > 0. Verify the inclusion ( F IY) c: r s{F\Y). Let a set Ae(F\Y) be given. Of course, it has the form A = Fn Y , where F e F . Since F cz r s(F), there is a set F ' e F with r s(F') cz F. So we have r s(F' n Y) cz F n Y = A, i.e., A e T s( F | Y). The other facts are obvious.

If F eF(X) satisfies the condition F cz r s{F), where s > 0, then we say that F is an s-modular filter (without property (bM2); cf. with [9]).

2.3.3. Fix a number s > 0 and let a filter F eF(X) have a base consisting of absolutely s-convex sets. Then F is called a locally s-convex modular filter (without property (bM2)). Obviously, this filter is s-modular without property (bM2). It is easy to see that F\Y is a locally s-convex modular filter with property (bM2). Namely, if 01 is a base of F consisting of absolutely s-convex sets, then the family X%\ Y:= {B n Y: BeX%) is a base of the filter F\Y and consists of absolutely s-convex sets.

2.3.4. Let a filter F e F ( X ) satisfy condition (bMl) with a certain fie К having \ц\ < 2 . Then F\Y is an LT-filter.

3.1. Give some lemmas on modular filters of a fixed character y > 0.

3.1.1. Let filters 3FjeF(X) (je J ф 0) he modular of the character у (with property (bM2)). Then the filter sup^ jEF(X) (constructed as an element of

je J

F(X)) exists and is modular of the character y.

Proof. For every j e J the inclusion ^ c= [0] holds (i.e., if Fg#^-, then OgF). Therefore s u p ^ e F (X ) exists ([3], [1]). Let := s u p W e shall

je J je J

show that 3F с= - A{ ^) . Take a set F e # -. Then there are Flf F

У ^{je J}

satisfying the equality F1 n ... n F n = F . Moreover, we can find sets F\, F'n

eU & j such that

je J

3. Properties o f modular filters.

Fl =>-A(F'i) , . . . , F , = , - â ( F n).

y y

Therefore

F - A (Fi) n . . . n -А (F'n) => -A (Fi n ... n F’„) g -A {3F).

y y y y

Thus F e- A ( ^ ) . Now suppose that an element x eX is given. Since c Vx

у J

for each j e J , the inclusion ^ a Vx holds.

3.1.2. If 3FjEF(X) (/gJ # 0 ) are LF-filters, then sup#-^ exists (as an

je J

element of F(X)) and is an LT-filter.

3.1.3. Let filters $F x, ..., SFn eF(X) be modular of the character y. Then the filter x + ... is also modular of the character y.

Proof. Let us write ^ : = & k + ... + & n. First prove that 3F c= - A(,$F).

1 Consider a set Fe$F. We can choose sets Ft g^ , ..., F„g such that F =з Fj + ... + Fn. Moreover,

Fk zD-A(F'k) (k — l , . . . , n) for some sets F\e^ F ' „ e3Fn.

Therefore, we have

F = -(A (F\) + ... + Л TO) = -A (F, + ... + F'n) e - d TO,

У ’ У У

and hence

Now show that c Vx for all x eX. Take an element x eX. We get

& = + ... + c;

# '1

c= Vx; so c= Fx.

3.1.4. Suppose that for к = 1, ..., n a filter fFkEF(X) is modular of a character yfe > 0. Then the filter A+ ... + #"„ is modular of the character max(y

l5

..., y„).

This is a corollary of 3.1.3 and 1.3.4.

3.1.5. Suppose that a filter ïF eF (X) is modular of the character y and let AeK, А ф 0. Then the filter A3F is also modular of the character y.

Proof. Since y#- <= A{fF), we have y(A^F) c= A(A£F). Now take an element x e X . Of course, ^ cz Vx; therefore, A3F c= AVx = Vx.

3.2. Fix a number s > 0 and give similar lemmas on s-modular filters.

3.2.1. If filters 3FjeF(X), where j e J Ф&, are s-modular, then the filter sup 3F ■ exists and is s-modular.

je J

Proof. Let filters ^ jE F (X) {j eJ Ф0) be s-modular. They are modular of the character

2

1/s, so the filter 3* := supJ^ exists and is modular of the

je J

character 21/s. Therefore, it is enough to show that c= Г8(^). Consider a set Fe$F. It has the form F — F1 n ... n Fn, where Flt ..., F„e[J 3F}. Moreover,

jeJ

there exist sets F\, ..., F'nE (J 3F-} such that f t э r,(F i)... Fn zdr s(F'„).

je J

Hence, we have F zd r s(F\) n ... n r s(F'„) => r s(F\ n ... n F'n). Furthermore, F

1

n . . . n F 'e # '; therefore Т е Г 5(^).

3.2.2. Assume that filters ..., „eF(X) are s-modular. Then the filter ,fF : = Jir1+ ... + # „ is also s-modular.

Proof. Since the filter 3F is modular of the character 21/s, we need only prove that fF с Г8(^). Let a set FeïF be given. Take sets Fxe3F x, ..., FnE ^ n for which F zd Ft + ... + F„. Moreover, choose F\

e3F ..., F'„e satisfying the conditions: Ft zd Fs(Fi), ..., Fn zd r s(F'n).

Hence we have F =э r s(F\)+ ... + r s(F'n) zd r s(F\+ ... +F'„). Verify the last inclusion. Consider an element xETfF'x + . . . +F'n). It has the form x = a(x1+ . . . + x n) + P{y1+ . . . + y n), where x1? y t E F\ ,..., x„, ynE F'„, а, @еК, |a|s + |/?|s <

1

; so we have

x = (ccXj^Apyfj-f ... + (ax„ + /?y„)*eFs(F'i) + ... + f s(F'n).

Finally, we can see that F e /y j^ ).

3.2.3. If for к = 1, ..., n a filter FueF{X) is sk-modular (sk > 0) and s0: = min(s1( ..., s„), then the filter F x + ... + F n is s0-modular.

3.2.4. Let XeK, X ф 0, and suppose that a filter F eF{X) is s-modular.

Then the filter XF is ilso s-modular.

Proof. Observe that r s(XA) = ХГ3(А) for A cz X.

3.3. Let s be a positive number.

3.3.1. Suppose that FjEF(X), where j E J ^ 0 , is a locally s-convex modular filter. Then the supremum sup F jE F (X) exists and is a locally s-convex

jeJ modular filter.

Proof. Let F j — [_&j] (/eJ), where is a locally s-convex modular base. By virtue of 1.3.11 and 3.2.1 we infer that 3F \= sup F j exists and is an s-modular filter. Observe that jeJ

ueN; Bx, ..., B„e (J &j]

je J

is a locally s-convex modular base and F = [_Щ.

3.3.2. If F x, ..., F nEF(X) are locally s-convex modular filters, then F x+ ... + F n is also a locally s-convex modular filter.

Proof. It is enough to observe that if ..., 0&n are locally s-convex modular bases in X, then the family

{Bx+ ... + B n: Bxe£%x, ..., BnE&n]

is also a locally s-convex modular base in X.

3.3.3. If F kEF(X) is a locally sk-convex modular filter, sk >

0

, к = 1, ..., n, then the filter F x + ...

4

- F n is a locally s0-convex modular filter, where s0: = min(Sj, ..., s„).

3.3.4. If F e F (X) is a locally s-convex modular filter and ХеК, X ф 0, then XF is also a locally s-convex modular filter.

3.4. We now intend to show that the families FMy(X), FSM(X), FMS.C (X) (y >

0

, s >

0

) are complete lattices ([

2

], [

6

]).

3.4.1. Let у > 0 and let H c FMy(X), H Ф 0. We have proved that supF(X)H (the supremum constructed as an element of F(X)) exists and supF(X)HeFM7{X). Therefore supF(X)H = supFMy(X)H (the symbol supFMy(X)H means that the supremum is constructed as an element of the set FMy(X); of course, it is constructed with respect to the inclusion relation). Moreover,

observe that infFM^X)FMy(X) exists and

mlFMy(x)FMy(X) = { * } = infnx,FM,(A-) = ibîr{x,F{X)

(the filter {X} is modular of the character

7

; furthermore, it is a locally s-convex modular filter for any s > 0). Hence we see that the family FMy(X)

(7

> 0) is a complete lattice; for H <= FMy(X), H Ф 0 , we have

infFMy(X)H = supFMy(X){ F e F M y(X): F c ^ for ev ry Уе Н)

= supP(X){ F e FMy(X): F а У îov ead ^ е Я }.

Moreover,

inffm7(X)0 = SUP FMytx)FMy(X) and

supFM^w 0 = infFM^(X)FMy(X) = {X}.

3.4.2. The families FSM(X) and FMS_C (X) (s > 0) are also complete lattices.

3.4.3. Investigate the following problem: Let

7

> 0 and let H c= FMy(X), H Ф 0 . Is then infF(X)H a modular filter? Analogous problems can be considered for the families FSM(X) (s > 0) and FMS_C(X) (se(0, 1».

Let us give an adequate example. X : = R x R over R,

F : = [{( —

6

, s) x R: e > 0 } ] , 0 := [ { Rx( — e, r.): 8 > 0 }].

Observe that F , ^ are locally convex LT-filters and

F глУ = infFW{# ', У} = [{((-£ , e) х Я )и (Я х (-е , e)): e > 0}], I (F n%) = F 1(3? n <&) + I(F n <g) = { RxR} .

It is obvious that no number

7

> 0 can satisfy the condition

7

( F пУ) c= A( F n ^). The filters F > ^ are modular of the character

7

for any

7

> 0. Moreover, they are locally s-convex modular filters for each sg(0, 1).

However, the filter infFm{ ^ , is not modular.

3.4.4. Let us consider filters F ..., F FMy(X)

(7

> 0). Construct the filter F : = F x + ... + F n. Moreover, let ^ be the infimum F x л ... л FF n mfFM (X){ F

l5

..., Then F e F My(X) and F cz F ..., Therefore c= Since the filter ... (n components) belongs to t^(0), we get

^ + ... + ^ => № with some number Я ф 0. Furthermore, F => ^ + ... + so F ХУ. Hence we have Я^ c= i.e., ~ Observe, if

7

g (0, 2) (this means that FMy(X) = FLT(X); so __ , ^ are LT-filters), then ХУ = ^ for any Я # 0, and therefore F = In general, this equality need not be satisfied. Give an appropriate example. X: = R,

7

: = 2, n: =2, F l :=[I~\, : = [27]. Of course, F 2gFM2(X) (moreover, J

^2

eFM^XX)).

Furthermore, we have F x л F 2 = J

^2

and F X+ F 2 = [37].

4. Generating modular filters and modular pseudotopologies.

4.1. Let y be a positive number.

4.1.1. Suppose that F eF(X). Then there exists a unique filter &EFMy(X) such that У cz F and Ж a for every Ж e F My(X), Ж <= F .

Proof. Consider the family Ж := { Ж e FMy(X): Ж cr F ) . This family is non-empty because {Xjestf. Construct the filter ^ : = supF(X)«s/ (= supFM дал/).

It is obvious that У e F M y(X), a F and Ж <= ^ for all Ж Esé. У The filter ^ constructed above will be denoted by fmy(F).

4.1.2. Suppose that F x, F 2eF(X) and F X< F 2. Then the relation fmy{ F x) < f m y{ F 2) holds.

Proof. Let (SX : = fmy( F J and У2\ = fmy( F 2). Since F x F 2, we have v.F x c= F 2 for some number а 0. Therefore x c= ci J^2, and con­

sequently c= Ж2. Of course, e F M y(X); hence c= ^ 2, i.e., ^ -< ^ 2.

4.1.3. If F

l5

J*

2

eF(X) and ~ ^ 2, t/zen fmy( F x) ~ fmy( F 2).

4.1.4. Let & : = fmy(F), x:=x<# and suppose that F eo(0), where the pseudotopology о is modular of the character y. Then F ex(0) and a < t.

Proof. Since => 0 , we have ^ е т ф ). Assume that Ж e F My(X) and о = т^. The condition F e o(0)implies ХЖ c for a certain number Я # 0.

Therefore c= c§\ so ^ x$, i.e., сг ^ t.

4.1.5. Let and f > y . Then fmp(fmy(F)) = f my{fmp(F)) = fmy(F). Generally, if F eF(X), yx, ..., yn > 0 and y0: = т т ( у 1? ..., y„), then fmyn( .. Jmy2(fmyi ( F ))... ) = fmyo(F).

Proof. The equality fmp(fmy(F)) = fmy(F) is obvious because fmy{F) e F Mp(X) (1.3.4). Let &: = fmfi(F), Ж: = № у(&) and Jf: = fmy{F). Then we have Ж ^ з Ж , Ж e FMy(X), Ж e FMy(X) a FMp(X). Since Ж с Ж , Ж e FMу(Х) and Ж = fmy(F), we get Ж c= Ж . From the conditions Ж e FMp(X), Ж c= F , У — fmp(F) we obtain the one: Ж с У . Moreover, Ж = fmy(&) and ЖеРМу(Х), ^{s o} jT ^{c z} Ж . Finally, Ж = Ж , i.e.,fmy(fmp{F))

= fmy(F).

4.1.6. Let F eF(X). Then fmx{F) is an LT-filter. We write flt(F) : = f mx(F). If a pseudotopology x eLP(X) satisfies condition (M) with a filter F , then F ~ : = flt(F ) is the neighbourhood filter o / O e l for the Wiweger topology

4.1.7. Suppose that a filter F e F(X) satisfies the conditions:

(2bMl) I F => ^F with a certain number Ç ф 0,

(bM2) V xx^F for every x eX.

Then

= [{ U ( F ,+ + F„): F „ F 2> ...e # - } ].

n=

1

P r o o f (cf. with [7]). It is obvious that the family &: = { (J (Fl + ... + F n):

n =

1

F

l5

F2, . . .еЖ] is a filter-base in X. We are going to show that := \0F\

eFLTfX). Verify the appropriate conditions:

(LT1) Ж си ТГ + ТГ,

(LT2) тГ <п /тГ,

(LT3) тГ с= Кх for every х е Х .

We have

00 00

U ((^i n ^2)+ "h(F2k — i n F2k))+ IJ ((^i n F 2)+ ... +(F2k- 1 n F a ))

fc=i fc=i

00

<= U ( f i + - + F . ) n= 1

for F1? F2, .. .g# -; therefore condition (LT1) is satisfied.

Verify (LT2). Let v:= 1/|£|. Then, by virtue of condition (2bMl), the inclusion 3F c= v/J* holds. Take sets F1, F 2, . . . е Ж and choose F\, F'2, .. .g3F such that F1 и» vIF\, F2 => vIF'2, ... We get

00 00 00

и (F

1

+ ... +F„) = U (v/Fi + ... +vIF’n) z> v/ U (F i+ ... +Fi).

n=l n=l n

=1

Thus У" си v/iF\ From (LTl)we obtain iF" си 2Ж. Therefore fF си

2

"iF' for each

hgN. Take an neN such that 2 " v ^ l. Then Ж си у\Ж <и \12пЖ a I'V (condition (LT2) is satisfied). Take an element x e X . We have & c Kx (condition (bM2)). Moreover, си J* ; therefore тГ c Kx, and (LT3) holds. Let JfeFLTiX), Ж а р . Then

GO

# = [{ U ( я , + ... +Я„): Я „ Н2, . . . eJfj] с Г п =

1

(cf. with [1], 5.4.8).

4.1.8. Let SF eF(X) and suppose that every set of the filter Ж is absorbent in X. Then

M & ) = [{ U (G ,+ ... +G„): G „ G2> ...e /iF } ] n=

1

00

= [ { U U F 1 + ...+ /F „ ): Fl t F2, . . . e ? } l n=

1

Proof. It is obvious that the filter l Ж satisfies condition (2bMl): 1{1Ж) = 1Ж. Since 1Ж ^ Ж , every set of 1Ж is absorbent in X.

Therefore ([1], 5.3.4) the filter 1Ж satisfies condition (bM2): Vx =э 1Ж for x e X . Next, by virtue of 4.1.7, we have

00

jlt(ISF) = [ { U (G 1 + . . . + G „ ) : G , , G 2, . . . 6 / ^ } ] /

1=1

00

= [{ U ( I F ^ . - . + I F J : F l t F

2

, . . . e. W}].

n=

1

The inclusion 1Ж c Ж implies flt(I^F) c flt(Ж). Since ]и{Ж) c Ж, we get flt(Ж) = Щ{Ж) c: therefore /к{Ж) c= /к(ТЖ). Finally, ]Ь(Ж) = /к(1Ж).

4.1.9. We have also proved that /к(1Ж) = /к(Ж) for ЖеР(Х).

4.1.10. Let H œ FMy(X), Н Ф 0 . Then

fmy(C

1

H) = infFMy(X)H, i.e., fmy(mîF(X)H) = infFMyWH.

Proof. For every # е Я we have /ш?( Р ) Я ) с Р ) Я с # ; therefore /my(P) Я) c inf^jvf (Х)Я. On the other hand, for ЖeH the mansion infFM {X)H c J5- holds. Hence infFM (X)H a f ) H, and consequently infFM (X)H

czfmy(f]H).

4.1.11. Let Я c FLT(X), H Ф 0 . Then У?г(П 7/) = inUlt(X)H',

here the filterflt(f) H) can be constructed in the manner given in 4.1.7 (the filter f ] H is balanced and f ] H czVx for every xeX) .

4.2. Let s be a positive number.

4.2.1. Suppose that ЖеР(Х). Then there exists a unique filter f sm(,Ж) e F sM( X) such that /8т(Ж) c= Ж and Ж <^fsm(<Ж) for every Ж e FsM(X), Ж c; Ж . Moreover, we have

W # ) = supFI„ { J f s F sM(X): / c f }

( = ырг,М(х){ Ж e F sM(X): Ж c :F\).

Observe that the family { Ж e F SM(X): Ж c f } is non-empty.

4.2.2. If Ж x, Ж2еР(Х), Ж x < Ж 2 {Ж x ~ Ж2), then /хт{Ж х)< / 3т(Ж 2)

4.2.3. Suppose that Ж e F( X) and У := /вт{Ж). Let a pseudotopology (teLP(X) be s-modular and let Жеа(0). Then Жет^(0) and a < x9.

4.2.4. Suppose that Jr eF(X), sx, s n> 0, s0:= max(

5l5

. . . , s n). Then fsnm( .. Js2m (fSl m{3F) ) ... ) = / 5от ( ^ ) .

4.2.5. Let &^eF{ X). Then f sm{13?) =

Proof. We have f sm{I3F) c= lgF c: therefore f sm(I^) c:f m ^ ) . On the other hand, / sm(^) = I f m ^ ) c /#"; hence / sm(#") c= fsm(LF).

4.2.6. Let H с FSM(X), H Ф 0 . Then / Sm(H H ) ~ ÛrfFSM(X)H .

4.3. Of course, one can also generate locally s-convex modular filters (s > 0). Analogs of Theorems 4.2.1-4.2.5 are true; we shall not formulate them.

Let the symbol /ms_c.(# ”) denote the locally s-convex modular filter generated by ^ eF(X) (s> 0). We have

Ms-cX^) = supF(x} {У eFMS.C (X): У C J5"}.

4.3.1. Suppose that 3F eF(X), ^s> 0 and let every set of 3F he absorbent in X. Then

a b s c o n v ^ ^ [{absconvsF: F e # "}] = ftns-cX^r)-

Proof. It is obvious that {absconvsF: Fe^ } eB(X). Since absconvJJr c= every set of abscony,#- is absorbent in X; therefore abscony,^

eFMS_C (X). The condition absconVj.#' <= implies absconvs#^ c= fms.c (J*).

On the other hand, from the inclusion fms.c {&) c we get /Щ-сХ^) = absconvs(/ms_c.(^)) <= absconvs^ .

4.3.2. Let s > 0 and suppose that H c= FMS.CXX), H ф 0 . Then

М -с .( П я ) = infFMs.c.(x)#;

moreover, using 4.3.1, we have

absconvs(P) # ) = in fFMs_c(X)H.

5. Lattices of modular pseudotopologies. Let us mention a certain general theorem on linear pseudotopologies. First recall ([1]) that for x

l5

x2eLP(X) the symbol ^ t

2

means that x^O) => t

2

(

0

).

Suppose that {xj}jeJ c= LP(X), J Ф 0 . Then infx;, supXjELP(X) exist and

je J je J

(infXj)(0) = ( ^eF(X): for some ..., &„e(J x/O)},

je J je J

(supx

7

)(0)= P)x

7

(0). So LP(X) is a complete lattice.

je J je J

2 — Comment. Math. 29.1

The proof is simple. Let

t(x) : = { J* eF{ X) :

3

^{F ^ + ...} for certain

^ l5

&ne\ Jxj (

0

^{j} + x,}

je J

x e X . It is easy to show that xeLP(X). Verify, for example, conditions (1) and (P) of [1]. Suppose that 3F, ^ei(O) and let & Ъ Уx + ... + ^ m, У => ^ m +

1

+ ... + &„, where ^ + . . . + ^ „e (J т

7

(

0

). Then + ^ <^ l + ... +^„; therefore ЗР+ У

je J

ei(0) (condition (1) is satisfied). On the other hand, for the same filters

^ , . . . , we have

[0])+ n [0]) = («?, n [0])+ ... + (» „ n [0]), ï = (Sf

1

n [ 0 ] ) + . . . + ( » „ n [

0

]); * , n [ 0 ] ... » „ n [0]

6

(J т,.(0).

je J

Hence ^ п ^ э ^

1

п [ 0 ] + . . . + ^ п [ 0 ] , i.e., SF слУехЩ (condition (p) is satisfied). It is obvious that т ^ x-} for j e J . Suppose that aeLP(X), a ^ т j for j e J and take a filter «^ет(О). Then we have ту(0) c er(0) for j e J (i.e., (J тДО) c= (j(0)) and 3F => + ... + ^ „ , where ^ 1? ..., <3 n are certain filters of

je J

(J т^.(О) cz <r(0). So =эfSx + ... + £ <r(0), and therefore £ o-(0). Finally, we

je J

get т(0) c ct(0), i.e., а ^ x. The second part of the proof is also easy.

5.1. Let the symbol LT(X) denote the set of all linear topologies on X. Of course, we have LT(X) c LP(X).

5.1.1. Suppose that H c. LT(X), Н Ф 0 . Then supLP{X)H eLT(X) (and therefore swpLP(X)H = supLT(X)H).

P r o o f (see also [5]). Let H = {xj}jsJ, where xjeLT(X) for j e J and J Ф 0 . Suppose that xj — x^j with & jEFLT(X), j e J. Moreover, let x:=supLP(X)H and # ”:= sup^JF., By virtue of 3.1.2 we deduce that

je J

3F eFLT(X). Furthermore, we have T(0) = O / 0 )

je J

= p) {%eF(X): У => &j) = {9eF{ X) : У гэ & . for j e J}

je J

= {&eF(X): & з #■} = TjF(

0

^).

Therefore xeLT(X).

5.1.2. Consider the anti-discrete topology x{X) and observe that x{X) < x for

every teLT(X). Therefore x{X} = mïLT{X)LT(X) (= inïLP{X)LP(X)) and for H cz LT(X) we have

inflt(x)H — supLP{X) {a e LT(X): a < т for every теЯ }

= supLT(X) {a ^eLT(X): a ^ т for every xeH).

Thus we obtain the well-known result: the set LT(X) is a complete lattice.

5.2. Suppose that у > 0 and let PMy(X): = {x^: 2F e F My(X)}.

5.2.1. Assume that & , ^ e F M y(X) and consider the pseudotopologies x& a Xcg : = т^р(Х){т^, Xc§\, x^ v x^:= s\xpLP^ {x ^ , x#}.

We have

Tjf A T eg = X p + <g£ PM y(X\ Xp V X(g — T^p v eg G PMy(X),

where Ж v У \ = supFW{ ^ , ^ }. Therefore the set PMy(X) is a sublattice of the lattice LP(X). Moreover, PMy(X) has the smallest element and the largest element.

Proof. Observe that Ж + Ж v У e F My(X) (3.1.3, 3.1.1). Let Ж e(xp л %)(0). Then Ж zd X2F + рУ with some numbers X, p Ф 0. Furthermore, we have

ХЖ + р№ zd XIЖ + p№ zd (|A| + \p\)FF + (|A| + M) 1У ZD (|A| + M)(A (Ж) + А (Щ ZD (\X\ + \ii\)y{3? + <S).

Therefore Ж ех^ + ^(0). On the other hand, if Р( Х) э Ж zd Х(Ж+ У) (i.e., Же х ^ + #(0)), then Же ( х^ л т^)(0). Hence we get (т^ л т^)(0) = v +ÿ(0).

Thus Tj*r л x<g = хр + # е Р Му(Х).

Now let Же ( хр v xf}(0). Then n t^(0). So Ж zd X Ж zd pfg with some numbers X, рфО. We have

X3F zd Х1Ж zd (\Х\ + \р\)1Ж zd (\X\A\p\)y^;

therefore X2F zd (|A| + \р\)уЖ. Similarly, p& zd (|A| + \p\)y<&. The above inclusions imply that

1

P + N)yЖ zd & and ¹

(\M + M)yж ^ д . So we get

1 -Ж zd & v <§, i.e., Ж zd (|A| + \p\)y{& v У).

(W + M)y

Hence Ж е т ^ у#(0). Next,; verify the inclusion т^у^(0) с (t^v't^KO). Let

a filter Ж е Tjrv^(O) be given. Then Ж -=> Х{Ж v with some number Я Ф 0.

Therefore we get => Я#-, => Я^, i.e., Ж е x^r(O) n x^(0) = (x^ v x^)(0). We have proved that x^ v x<$ = xjrv^. Of course, x^ v#e PMy(X).

Observe that тт еРМ у(Т) and xw ^ x for every x e P My(X). Thus the topology x{X} is the smallest element of PMy(X).

Let Ж gFMy(X). Construct the filter J* v : = У Ж . Then Ж c: v (because / “ ех^(0)) and x^ ^ x ^ = (x^Y (see [1], 5.4.5). Moreover, consider the topology x:= supLr(X){x ^ : gFMy(X)} = supLr(X)LT(X). For every &

e F My(X) we have x& ^ x^v ^ x e P My(X). Therefore x is the largest element of PMy(X).

5.2.2. Observe that for Ж, @ e F M y(X), y > 0 and XF л ^ : = infFM w {.^, we have x^ л x# = x^ ( s e e 3.4.4 and 1.2).

5.2.3. Consider the set (J PMy(X). Observe that for y

0

> 0 we have

7 > 0

(J PMy(X) = (J PMy(X) (see 1.3.4). It is easy to show that the set |J PMy(X)

У> Уо у > 0 7 > 0

is a sublattice of LP(X). Moreover, x{X] is the smallest element and suplt(X)LT(X) is the largest element of this sublattice.

5.3. Define PSM( X) : = { x ^ : & e F sM(X)}, where s ^> 0. One can show that the sets PSM(X) (s > 0), (J PSM( X) = (J PSM(X) (s

0

> 0) are sublat-

s > 0 se(0 ,so)

tices of the lattice LP(X). For Уе FsM(X) (s > 0) we have x& л x^ = x^+g, x& v x<§ — x&y<s. The topology x{X] is the smallest element and supLr(X)LT(X) is the largest element of these sublattices.

5.4. Let s > 0 and suppose that Ж, У e FMs_c (X). Then x^ л x# = x^ + g and xJr v x^ — Xjrv^.

5.4.1. We have shown that the set PMS.C(X): = {x^: Ж e F Ms_c (X)}

(s > 0 ) is a sublattice of the lattice LP(X). The set У PMS.C (X)

s > 0

= (J PMS_C'{X) (s0 > 0) is also a sublattice of LP(X). Observe that x{X) is the

se(0 ,so)

smallest element of these sublattices.

5.4.2. Fix a positive number s. Let ^ e F M s.9XX). Then # ' V:= F # '

= supF(X){ e^: £ > 0} ([1], 5.4.7). Of course, for e > 0 we have гЗР e FMs.c (X).

Moreover, supFW{e#': e > 0} = J*-" e F Ms.c (X) (3.3.1). Consider the filter

^ := supm ){ . r v: Ж e F Ms.c (X)}. Of course, e FLT(X) n FMS_C (X). For every 2F e F Ms.c (X) we have x# ^ ^ x^. Therefore хщ is the largest element of PMS_CXX).

5.4.3. Observe that we have proved the implication: if s > 0 and x ePM s.c (X), then x“ e P Ms.c (X) (obviously, xv eLT(X)).

Moreover, the following theorem is true:

Let s > 0 and let ie P M s.c.(X). Then x* e PMS_CXX) n LT(X).

Proof. It is enough to analyse the construction of the neighbourhood filter of 0 for ta (4.1.6, 4.1.7).

References

[1 ] E. A m b r o z k o , Pseudotopologies fo r modular spaces, Comment. Math. 25 (1985), 189-200.

[2 ] G . B ir k h o f f , Lattice Theory, M oscow 1984 (in Russian, translated from English).

[3 ] H . R. F is c h e r , Limesraume, Math. Ann. 137 (1959), 269-3 0 3.

[4 ] A. F r o l i c h e r , W . B u c h e r , Calculus in Vector Spaces without Norm, M oscow 1970 (in Russian, translated from English).

[5 ] W . G à h l e r , Grundstrukturen der Analysis II, Berlin 1978.

[6 ] G . G r â t z e r , General Lattice Theory, M oscow 1982 (in Russian, translated from English).

[7 ] R. L e s n i e w i c z , On generalized modular spaces I, Comment. M ath. 18 (1975), 2 23 -2 4 2.

[8 ] — , On generalized modular spaces II, ibidem 18 (1975), 243-271.

[9 ] — , W . O r l i c z , A N ote on Modular Spaces. X IV , Bull. Acad. Polon. Sci., Sér. Sci. Math.

Astronom. Phys. 22 (1974), 9 15 -9 2 3.