Proposition 18.

If

ν

ν

ν

1

then every neighborhood filter is a Cauchy filter, that it

ν

⊒ h

(

FCD

)

ν

i

{

x

} ×

RLD

h

(

FCD

)

ν

i

{

x

}

for every point

x

.

Proof.

h

(

FCD

)

ν

i

{

x

} ×

RLD

h

(

FCD

)

ν

i

{

x

}

=

h

(

FCD

)

ν

i↑

Ob

ν

{

x

} ×

RLD

h

(

FCD

)

ν

i↑

Ob

ν

{

x

}

=

ν

(

Ob

ν

{

x

} ×

RLD

Ob

ν

{

x

}

)

ν

1

=

ν

◦ ↑

RLD

(

Ob

ν

;

Ob

ν

)

{

(

x

;

x

)

}

ν

1

ν

id

RLD

(

Ob

ν

;

Ob

ν

)

ν

1

=

ν

ν

1

ν

.

Proposition 19.

If a filter converges to a point, it is a low filter, provided that every neighborhood

filter is a low filter.

Proof.

Let

F ⊑ h

(

FCD

)

ν

i

{

x

}

. Then

F ×

RLD

F ⊑ h

(

FCD

)

ν

i

{

x

} ×

RLD

h

(

FCD

)

ν

i

{

x

} ⊑

ν

.

Corollary 20.

If a filter converges to a point, it is a low filter, provided that

ν

ν

ν

1

.

6 Maximal Cauchy filters

Lemma 21.

Let

S

be a set of sets with

d

h↑

F

i

S

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

} ⊒ ↑

X

T

S

. Thus

h

S

T

S

T

i{

x

}

=

h

S

T

ih

S

T

i{

x

} ∈ h↑

FCD

S

T

i

d

h↑

F

i

S

F

{h↑

FCD

(

X

×

X

)

i

d

h↑

F

i

S

|

X

S

}

=

F

{↑

F

X

|

X

S

}

=

F

h↑

F

i

S

that is

h

S

T

S

T

i{

x

} ⊇

S

S

.

Corollary 22.

Let

S

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

T

=

{X ×

RLD

X | X ∈

S

}

Then

G

T

G

T

=

G

S

×

RLD

G

S.

Proof.

F

T

F

T

=

d

{↑

F

(

X

X

)

|

X

F

T

}

.

If

X

F

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

F

T

F

T

=

d

F

S

Q

T

P

Q

×

S

Q

T

P

Q

|

X

F

T

F

S

×

RLD

F

S

.

F

T

F

T

F

S

×

RLD

F

S

is obvious.

Definition 23.

I call an endoreloid

ν

symmetrically transitive

iff for every symmetric endofuncoid

f

FCD

(

Ob

ν

;

Ob

ν

)

we have

f

ν

f

f

ν

.

Obvious 24.

It is symmetrically transitive if at least one of the following holds:

1.

ν

ν

ν

;

2.

ν

ν

1

ν

;

3.

ν

1

ν

ν

.

4.

ν

1

ν

1

ν

.

Corollary 25.

Every uniform space is symmetrically transitive.

Maximal Cauchy filters

3