background image

Suppose for the contrary that

A

is innite. Then

A

contains more than one non-zero points

y

,

z

(

y

=

/

z

). Without loss of generality

y < z

. So we have that

(

y

;

z

)

is not of the form

(

y

;

y

)

nor

(0;

y

)

nor

(

y

; 0)

. Therefore

A

A

isn't a subset of

¡

.

12.2 Totally bounded endoreloids

The below is a straightforward generalization of the customary denition of totally bounded sets
on uniform spaces (it's proved below that for uniform spaces the below denitions are equivalent).

Denition 12.9.

An endoreloid

f

is

-totally bounded

(totBound

(

f

)

) if every

E

2

xyGR

f

is

-

thick.

Denition 12.10.

An endoreloid

f

is

-totally bounded

(totBound

(

f

)

) if every

E

2

xyGR

f

is

-thick.

Remark 12.11.

We could rewrite the above denitions in a more algebraic way like xyGR

f

thick

(with thick

would be dened as a set rather than as a predicate), but we don't really need

this simplication.

Proposition 12.12.

If an endoreloid is

-totally bounded then it is

-totally bounded.

Proof.

Because thick

(

E

)

)

thick

(

E

)

.

Proposition 12.13.

If an endoreloid

f

is reexive and Ob

f

is nite then

f

is both

-totally

bounded and

-totally bounded.

Proof.

It enough to prove that

f

is

-totally bounded. Really, every

E

2

xyGR

f

is reexive.

Thus

f

x

g  f

x

E

for

x

2

Ob

f

and thus

ff

x

g j

x

2

Ob

f

g

is a sought for nite cover of Ob

f

.

Obvious 12.14.

A principal endoreloid induced by a Rel-morphism

E

is

-totally bounded i

E

is

-thick.

A principal endoreloid induced by a Rel-morphism

E

is

-totally bounded i

E

is

-thick.

Example 12.15.

There is a

-totally bounded endoreloid which is not

-totally bounded.

Proof.

It follows from the example above and properties of principal endoreloids.

12.3 Special case of uniform spaces

Denition 12.16.

Uniform space

is essentially the same as symmetric, reexive and transitive

endoreloid.

Exercise 12.1.

Prove that it is essentially the same as the standard denition of a uniform space (see Wikipedia

or PlanetMath).

Theorem 12.17.

Let

f

be such a endoreloid that

f

f

¡

1

v

f

. Then

f

is

-totally bounded i it

is

-totally bounded.

Proof.

)

.

Proved above.

(

.

For every

"

2

GR

f

we have that

h

"

if

c

0

g

; :::;

h

"

if

c

n

g

covers the space.

h

"

if

c

i

g  h

"

if

c

i

"

"

¡

1

because for

x

2 h

"

if

c

i

g

(the same as

c

i

2 h

"

¡

1

if

x

g

) we have

hh

"

if

c

i

g  h

"

if

c

i

gif

x

g

=

h

"

if

c

i

gh

"

ih

"

¡

1

if

x

g

=

h

"

"

¡

1

if

x

g

. For every

"

0

2

GR

f

exists

"

2

GR

f

such that

"

"

¡

1

"

0

because

f

f

¡

1

v

f

. Thus for every

"

0

we have

h

"

if

c

i

g  h

"

if

c

i

"

0

and so

h

"

if

c

0

g

; :::;

h

"

if

c

n

g

:

154

Total boundness of reloids