background image

15. REST

60

Because

D

was taken arbitrary, we have

a

×

RLD

a

v

ν

that is

a

is

Cauchy.

. Suppose that Cauchy space associated with a reloid

ν

is totally bounded but

the reloid

ν

isn’t totally bounded. So there exists a

D

GR

ν

such that

(Ob

ν

)

\ h

D

i

F

6

=

for every finite set

F

.

Consider the filter base

S

=

(Ob

ν

)

\ h

D

i

F

F

P

Ob

ν, F

is finite

and the filter

F

=

d

h↑

Ob

ν

i

S

generated by this base. The filter

F

is

proper because intersection

P

Q

S

for every

P, Q

S

and

/

S

.

Thus there exists a Cauchy (for our Cauchy space) filter

X v F

that is

X ×

RLD

X v

ν

.

Thus there exists

M

∈ X

such that

M

×

M

D

. Let

F

be a finite

subset of Ob

ν

. Then (Ob

ν

)

\ h

D

i

F

∈ F w X

. Thus

M

6

(Ob

ν

)

\ h

D

i

F

and so there exists a point

x

M

((Ob

ν

)

\ h

D

i

F

).

h

M

×

M

i{

p

} ⊆ h

D

i{

x

}

for every

p

M

; thus

M

⊆ h

D

i{

x

}

.

So

M

⊆ h

D

i

(

F

∪ {

x

}

). But this means that

M

∈ X

does not intersect

(Ob

ν

)

\ h

D

i

(

F

∪ {

x

}

)

∈ F w X

, what is a contradiction (taken into

account that

X

is proper).

http://math.stackexchange.com/questions/104696/

pre-compactness-total-boundedness-and-cauchy-sequential-compactness

13. Totally bounded funcoids

Definition

2370

.

A funcoid

ν

is totally bounded iff

X

Ob

ν

∃X ∈

F

Ob

ν

: (0

6

=

X v↑

Ob

ν

X

∧ X ×

FCD

X v

ν

)

.

This can be rewritten in elementary terms (without using funcoidal product:

X ×

FCD

X v

ν

⇔ ∀

P

X

:

X v h

ν

i

P

⇔ ∀

P

X

, Q

X

:

P

[

ν

]

Q

P, Q

Ob

ν

: (

E

∈ X

: (

E

P

6

=

∅ ∧

E

Q

6

=

)

P

[

ν

]

Q

).

Note that probably I am the first person which has written the above formula

(for proximity spaces for instance) explicitly.

14. On principal low spaces

Definition

2371

.

A low space (

U,

C

) is

principal

when all filters in

C

are

principal.

Proposition

2372

.

Having fixed a set

U

, principal reflexive low spaces on

U

bijectively correspond to principal reflexive symmetric endoreloids on

U

.

Proof.

??

http://math.stackexchange.com/questions/701684/union-of-cartesian-

squares

15. Rest

https://en.wikipedia.org/wiki/Cauchy_filter#Cauchy_filters
https://en.wikipedia.org/wiki/Uniform_space

“Hausdorff completion of a uni-

form space” here)

http://at.yorku.ca/z/a/a/b/13.htm

: the category

Prox

of proximity spaces

and proximally continuous maps (i.e. maps preserving nearness between two sets)
is isomorphic to the category of totally bounded uniform spaces (and uniformly
continuous maps).