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12. TOTALLY BOUNDED

59

Proof.

Let

N

=

h

(

FCD

)

ν

i

{

x

}

. Let

C

w

N

be a Cauchy filter. We need to

show

N

w

C

.

Since

C

is Cauchy filter,

C

×

RLD

C

v

ν

. Since

C

w

N

we have

C

is a neigh-

borhood of

x

and thus

Ob

ν

{

x

} v

C

(reflexivity of

ν

). Thus

Ob

ν

{

x

} ×

RLD

C

v

C

×

RLD

C

and hence

Ob

ν

{

x

} ×

RLD

C

v

ν

;

C

v

im(

ν

|

Ob

ν

{

x

}

) =

h

(

FCD

)

ν

i

{

x

}

=

N

.

10. Cauchy continuous functions

Definition

2363

.

A function

f

:

U

V

is

Cauchy continuous

from a low

space (

U,

C

) to a low space (

V,

D

) when

∀X ∈

C

:

h↑

FCD

f

iX ∈

D

.

Proposition

2364

.

Let

f

be

a

principal

reloid.

Then

f

C((

RLD

)

Low

C

,

(

RLD

)

Low

D

) iff

f

is Cauchy continuous.

f

(

RLD

)

Low

C

f

1

v

(

RLD

)

Low

D

l

X ∈

C

(

f

(

X ×

RLD

X

)

f

1

)

v

(

RLD

)

Low

D

l

X ∈

C

(

h↑

FCD

f

iX ×

RLD

h↑

FCD

f

iX

)

v

(

RLD

)

Low

D

∀X ∈

C

:

h↑

FCD

f

iX ×

RLD

h↑

FCD

f

iX v

(

RLD

)

Low

D

∀X ∈

C

:

h↑

FCD

f

iX ∈

D

.

Thus we have expressed Cauchy properties through the algebra of reloids.

11. Cauchy-complete reloids

Definition

2365

.

An endoreloid

ν

is

Cauchy-complete

iff every low filter for

this reloid converges to a point.

Remark

2366

.

In my book [

2

]

complete reloid

means something different. I

will always prepend the word “Cauchy” to the word “complete” when meaning is
by the last definition.

https://en.wikipedia.org/wiki/Complete_uniform_space#Completeness

12. Totally bounded

http://ncatlab.org/nlab/show/Cauchy+space

Definition

2367

.

Low space is called

totally bounded

when every proper filter

contains a proper Cauchy filter.

Obvious

2368

.

A reloid

ν

is totally bounded iff

X

P

Ob

ν

∃X ∈

F

Ob

ν

: (

⊥ 6

=

X v↑

Ob

ν

X

∧ X ×

RLD

X v

ν

)

.

Theorem

2369

.

A symmetric transitive reloid is totally bounded iff its Cauchy

space is totally bounded.

Proof.

. Let

F

be a proper filter on Ob

ν

and let

a

atoms

F

. It’s enough to prove that

a

is Cauchy.

Let

D

GR

ν

. Let also

E

GR

ν

is symmetric and

E

E

D

.

There existsa finite subset

F

Ob

ν

such that

h

E

i

F

= Ob

ν

.

Then

obviously exists

x

F

such that

a

v↑

Ob

ν

h

E

i{

x

}

, but

h

E

i{

x

}×h

E

i{

x

}

=

E

1

(

{

x

} × {

x

}

)

E

D

, thus

a

×

RLD

a

v↑

RLD

(Ob

ν,

Ob

ν

)

D

.