9. MAXIMAL CAUCHY FILTERS

58

9. Maximal Cauchy filters

Lemma

2356

.

Let

S

be a set of sets with

d

F

S

6

= 0

F

(in other words,

S

has finite intersection property). Let

T

=

X

×

X

X

S

. Then

[

T

[

T

=

[

S

×

[

S.

Proof.

Let

x

S

S

. Then

x

X

for some

X

S

.

h

S

T

i{

x

} w↑

X

T

S

6

=

. Thus

h

S

T

S

T

i{

x

}

=

h

S

T

ih

S

T

i{

x

}

FCD

S

T

d

h↑

F

i

S

w

d

n

h↑

FCD

(

X

×

X

)

i

d

h↑

F

i

S

X

S

o

=

d

n

F

X

X

S

o

=

d

h↑

F

i

S

that is

h

S

T

S

T

i{

x

} ⊇

S

S

.

Corollary

2357

.

Let

S

be a set of filters (on some fixed set) with nonempty

meet. Let

T

=

X ×

RLD

X

X ∈

S

Then

l

T

l

T

=

l

S

×

RLD

l

S.

Proof.

d

T

d

T

=

d

n

F

(

X

X

)

X

d

T

o

.

If

X

d

T

then

X

=

S

Q

T

(

P

Q

×

P

Q

) where

P

Q

Q

. Therefore by the lemma

we have

[

P

Q

×

P

Q

Q

T

[

P

Q

×

P

Q

Q

T

=

[

Q

T

P

Q

×

[

Q

T

P

Q

.

Thus

X

X

=

S

Q

T

P

Q

×

S

Q

T

P

Q

.

Consequently

d

T

d

T

=

d

F

S

Q

T

P

Q

×

S

Q

T

P

Q

X

d

T

w

d

S

×

RLD

d

S

.

d

T

d

T

v

d

S

×

RLD

d

S

is obvious.

Definition

2358

.

I call an endoreloid

ν

symmetrically transitive

iff for every

symmetric endofuncoid

f

FCD

(Ob

ν,

Ob

ν

) we have

f

v

ν

f

f

v

ν

.

Obvious

2359

.

It is symmetrically transitive if at least one of the following

holds:

1

.

ν

ν

v

ν

;

2

.

ν

ν

1

v

ν

;

3

.

ν

1

ν

v

ν

.

4

.

ν

1

ν

1

v

ν

.

Corollary

2360

.

Every uniform space is symmetrically transitive.

Proposition

2361

.

(Low)

ν

is a completely Cauchy space for every symmetri-

cally transitive endoreloid

ν

.

Proof.

Suppose

S

P

n

X ∈

F

X ×

RLD

X v

ν

o

.

d

n

X ×

RLD

X

X ∈

S

o

v

ν

;

d

n

X ×

RLD

X

X ∈

S

o

d

n

X ×

RLD

X

X ∈

S

o

v

ν

;

d

S

×

RLD

d

S

v

ν

(taken

into account that

S

has nonempty meet). Thus

d

S

is Cauchy.

Proposition

2362

.

The neighbourhood filter

h

(

FCD

)

ν

i

{

x

}

of a point

x

Ob

ν

is a maximal Cauchy filter, if it is a Cauchy filter and

ν

is a reflexive reloid.

FiXme

: Does it holds for all low filters?