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1. THE REST

39

Proposition

2101

.

Let

g

be a reloid and

f

= (

FCD

)

g

and

f

=

f

f

1

. Then

f

×

in

f

1

RLD

w

g

.

Proof.

f

×

in

f

1

RLD

=

(

RLD

)

in

f

(

C

)

(

RLD

)

in

f

1

RLD

= (

RLD

)

in

f

1

RLD

(

RLD

)

in

f

1

= (

RLD

)

in

f

(

RLD

)

in

f

1

= (

RLD

)

in

(

f

f

1

) = (

RLD

)

in

f

=

(

RLD

)

in

(

FCD

)

g

w

g

.

Lemma

2102

.

Cor

h

f

×

(

A

)

f

i

g

v

1

RLD

if (

FCD

)

g

=

f

for a

T

1

-separable reloid

g

.

1.1. Propositions from

[

2

]

which do not hold for our products.

In

this subsection I present counter-examples against modified propositions from [

2

]

in which I replace Tychonoff product with our subatomic or cross-inner products.

TODO: Consider as a counter-example the non-transitive compact funcoid

n

(

x,y

)

x,y

[0;1]

,

|

x

y

|

<

1
3

o

.

Example

2103

.

1

Rel

×

(

A

)

1

Rel

1

Rel

A

1

Rel

.

Proof.

1

Rel

×

(

A

)

1

Rel

1

Rel

=

d

x

atoms 1

Rel

1

Rel

×

(

A

)

1

Rel

x

=

d

x

atoms 1

Rel

1

Rel

dom

x

×

RLD

1

Rel

im

x

=

d

x

atoms 1

Rel

dom

x

×

RLD

im

x

=

d

x

atoms

F

(

x

×

RLD

x

)

A

1

Rel

.

Statement 2 on page 172 of [

2

does not survive modification:

Example

2104

.

1

. There is a funcoid

f

and

V

up

f

such that

V

M

V

1

/

up

f

×

(

A

)

f

M

.

2

.

h

f

×

(

A

)

f

i

M

A

g

◦ ↑

RLD

M

g

1

for some reloid

g

, binary relation

M

and

the funcoid

f

= (

FCD

)

g

.

Proof.

1

Take

f

=

M

=

V

= 1

Rel

and use the example above.

2

Take

f

=

g

=

M

= 1

Rel

and use the example above.

Corollary

2105

.

h

f

×

(

A

)

f

i

M

v h

f

×

(

C

)

f

i

M

.

Corollary

2106

.

V

V

1

up

h

f

×

(

A

)

f

i

1

RLD

;

f

f

1

w h

f

×

(

A

)

f

i

1

RLD

.

Proof.

??

Remark

2107

.

I attempted to generalize the below theorem more than the

standard general topology theorem about correspondence of compact and uniform
spaces, but haven’t really succeeded much, as it appears to be needed that the reloid
in question is reflexive, symmetric, and transitive, that is just a uniform space as
in the standard general topology.

Does the reverse inequality hold, that is

g

w

f

×

(

A

)

f

1

RLD

and/or

g

w

f

×

in

f

1

RLD

(for compact

f

= (

FCD

)

g

)?

Theorem

2108

.

g

v

f

×

(

A

)

f

1

RLD

for compact

f

= (

FCD

)

g

. (We have

already proved this in an easier way, and not only for compact funcoids.)

Suppose there is

U

up

h

f

×

(

A

)

f

i

1

RLD

such that

U /

up

g

.

Then

n

V

\

U

V

up

g

o

=

g

\

U

would be a proper filter.

Thus by reflexivity

h

f

×

(

A

)

f

i

(

g

\

U

)

6

=

.

By compactness of

f

×

(

A

)

f

, Cor

h

f

×

(

A

)

f

i

(

g

\

U

)

6

=

.