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5. RELATIONSHIPS WITH SYMMETRIC RELOIDS

51

Proof.

Let

F

be a proper Cauchy filter. Then

d

{

X ∈

C

X wF

}

n

X ∈

C

X wF

o

(existing by

the above proposition) is the maximal Cauchy filter containing

F

.

Suppose another maximal Cauchy filter

T

contains

F

. Then

T ∈

n

X ∈

C

X wF

o

and

thus

T

=

d

{

X ∈

C

X wF

}

n

X ∈

C

X wF

o

.

5. Relationships with symmetric reloids

FiXme

: Also consider relationships with funcoids.

Definition

2311

.

Denote (

RLD

)

Low

(

U,

C

) =

d

n

X ×

RLD

X

X ∈

C

o

.

Definition

2312

.

(Low)

ν

(

low space

for endoreloid

ν

) is a low space on

U

such that

GR(Low)

ν

=

X ∈

F

(

U

)

X ×

RLD

X v

ν

.

Theorem

2313

.

If (

U,

C

) is a low space, then (

U,

C

) = (Low)(

RLD

)

Low

(

U,

C

).

Proof.

If

X ∈

C

then

X ×

RLD

X v

(

RLD

)

Low

(

U,

C

) and thus

X ∈

GR(Low)(

RLD

)

Low

(

U,

C

). Thus (

U,

C

)

v

(Low)(

RLD

)

Low

(

U,

C

).

Let’s prove (

U,

C

)

w

(Low)(

RLD

)

Low

(

U,

C

).

Let

A ∈

GR(Low)(

RLD

)

Low

(

U,

C

). We need to prove

A ∈

C

.

Really

A ×

RLD

A v

(

RLD

)

Low

(

U,

C

). It is enough to prove that

∃X ∈

C

:

A v

X

.

Suppose

@

X ∈

C

:

A v X

.

For every

X ∈

C

obtain

X

X

∈ X

such that

X

X

/

∈ A

(if forall

X

∈ X

we have

X

X

∈ A

, then

X w A

what is contrary to our supposition).

It is now enough to prove

A ×

RLD

A 6v

d

n

U

X

X

×

RLD

U

X

X

X ∈

C

o

.

Really,

d

n

U

X

X

×

RLD

U

X

X

X ∈

C

o

=

RLD

(

U,U

)

S

n

U

X

X

×

RLD

U

X

X

X ∈

C

o

.

So our claim

takes the form

S

n

U

X

X

×

RLD

U

X

X

X ∈

C

o

/

GR(

A ×

RLD

A

) that is

A

∈ A

:

S

n

U

X

X

×

RLD

U

X

X

X ∈

C

o

+

A

×

A

what is true because

X

X

+

A

for every

A

∈ A

.

Remark

2314

.

The last theorem does not hold with

X ×

FCD

X

instead of

X ×

RLD

X

(take

C

=

n

{

x

}

x

U

o

for an infinite set

U

as a counter-example).

Remark

2315

.

Not every symmetric reloid is in the form (

RLD

)

Low

(

U,

C

)

for some Cauchy space (

U,

C

).

The same Cauchy space can be induced by

different uniform spaces. See

http://math.stackexchange.com/questions/702182/

different-uniform-spaces-having-the-same-set-of-cauchy-filters

Proposition

2316

.

1

. (Low)

f

is reflexive iff endoreloid

f

is reflexive.

2

. (

RLD

)

Low

f

is reflexive iff low space

f

is reflexive.

Proof.

1

.

f

is reflexive

1

RLD

v

f

⇔ ∀

x

Ob

f

:

(

{

x

} ×{

x

}

)

v

f

⇔ ∀

x

Ob

f

:

{

x

RLD

↑ {

x

} v

f

⇔ ∀

x

Ob

f

:

↑ {

x

} ∈

(Low)

f

(Low)

f

is reflexive.

2

Let

f

is reflexive. Then

x

Ob

f

:

↑ {

x

} ∈

f

;

x

Ob

f

:

↑ {

x

RLD

{

x

} v

(

RLD

)

Low

f

;

x

Ob

f

:

(

{

x

} × {

x

}

)

v

(

RLD

)

Low

f

; 1

RLD

v

(

RLD

)

Low

f

.

Let now (

RLD

)

Low

f

be reflexive. Then

f

= (Low)(

RLD

)

Low

f

is reflexive.