7.

(

RLD

)

in

(

f

1

f

)

[TODO: Use this below.]

Proof.

??

Proposition 10.

Let

g

be a reloid and

f

= (

FCD

)

g

. Then

h

f

×

f

i

g

.

Proof.

h

f

×

f

i

RLD

Y

⇔ ↑

RLD

∆ [

f

×

f

]

RLD

Y

⇔ ↑

FCD

f

×

(

C

)

f

FCD

Y

f

◦ ↑

FCD

f

1

FCD

Y

f

f

1

FCD

Y

f

FCD

Y

f

⊓ ↑

FCD

Y

0

(

RLD

)

in

(

f

⊓ ↑

FCD

Y

)

0

(

RLD

)

in

f

(

RLD

)

in

FCD

Y

0

(

RLD

)

in

f

(

RLD

)

out

FCD

Y

0

(

RLD

)

in

f

⊓ ↑

RLD

Y

0

(

RLD

)

in

(

FCD

)

g

⊓ ↑

RLD

Y

0

g

⊓ ↑

RLD

Y

0

g

RLD

Y

.

Proposition 11.

Let

f

be a funcoid. Then

V

M

V

1

GR

h

f

×

f

i

M

for every

V

GR

f

.

Proof.

V

M

V

1

GR

(

f

◦ ↑

M

f

1

) =

GR

f

×

(

C

)

f

M

GR

h

f

×

f

i↑

M

=

GR

h

f

×

f

i

M

.

[FIXME: Wrong direction of

.]

Because

FCD

X

f

×

(

C

)

f

FCD

M

⇔ ↑

RLD

X

h

f

×

f

i↑

RLD

M

(

FCD

)(

RLD

X

⊓ h

f

×

f

i↑

RLD

M

)

0

(

FCD

)

RLD

X

(

FCD

)

h

f

×

f

i↑

RLD

M

0

(

FCD

)

RLD

X

(

FCD

)

h

f

×

f

i↑

RLD

M

⇔ ↑

FCD

X

(

FCD

)

h

f

×

f

i↑

RLD

M

;

f

×

(

C

)

f

FCD

M

(

FCD

)

h

f

×

f

i↑

RLD

M

GR

f

×

(

C

)

f

FCD

M

GR

(

FCD

)

h

f

×

f

i↑

RLD

M

GR

h

f

×

f

i↑

RLD

M

Proposition 12.

h

f

×

f

i

M

g

◦ ↑

RLD

M

g

1

whenever

(

FCD

)

g

=

f

for a reloid

g

.

Proof.

For every

V

GR

g

we have

V

M

V

1

GR

h

f

×

f

i

M

. Thus

g

◦ ↑

RLD

M

g

1

=

d

{

V

M

V

1

|

V

GR

g

} ⊒

d

GR

h

f

×

f

i

M

=

GR

h

f

×

f

i

M

.

Corollary 13.

h

f

×

f

i

M

f

×

(

C

)

f

M

.

Corollary 14.

V

V

1

GR

h

f

×

f

i

;

f

f

1

⊒ h

f

×

f

i

.

Proof.

??

Lemma 15.

Cor

h

f

×

f

i

g

if

(

FCD

)

g

=

f

where

(

FCD

)

g

=

f

for a

T

1

-separable reloid

g

.

Proof.

??

Remark 16.

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.

Theorem 17.

Let

f

be a

T

1

-separable compact reflexive symmetric funcoid and

g

be a reloid such

that

1.

(

FCD

)

g

=

f

;

2.

g

g

1

g

.

3