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13.3. SPECIAL CASE OF UNIFORM SPACES

233

To prove that Γ is not

α

-thick it’s enough to prove that every set

A

such that

A

×

A

Γ is finite.

Suppose for the contrary that

A

is infinite. Then

A

contains more than one

non-zero points

y

,

z

(

y

6

=

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 Γ.

13.2. Totally bounded endoreloids

The below is a straightforward generalization of the customary definition of

totally bounded sets on uniform spaces (it’s proved below that for uniform spaces

the below definitions are equivalent).

Definition

1239

.

An endoreloid

f

is

α

-totally bounded

(totBound

α

(

f

)) if ev-

ery

E

up

f

is

α

-thick.

Definition

1240

.

An endoreloid

f

is

β

-totally bounded

(totBound

β

(

f

)) if ev-

ery

E

up

f

is

β

-thick.

Remark

1241

.

We could rewrite the above definitions in a more algebraic way

like up

f

thick

α

(with thick

α

would be defined as a set rather than as a predicate),

but we don’t really need this simplification.

Proposition

1242

.

If an endoreloid is

α

-totally bounded then it is

β

-totally

bounded.

Proof.

Because thick

α

(

E

)

thick

β

(

E

).

Proposition

1243

.

If an endoreloid

f

is reflexive and Ob

f

is finite then

f

is

both

α

-totally bounded and

β

-totally bounded.

Proof.

It enough to prove that

f

is

α

-totally bounded. Really, every

E

up

f

is reflexive. Thus

{

x

} × {

x

} ⊆

GR

E

for

x

Ob

f

and thus

n

{

x

}

x

Ob

f

o

is a sought

for finite cover of Ob

f

.

Obvious

1244

.

A principal endoreloid induced by a

Rel

-morphism

E

is

α

-totally bounded

iff

E

is

α

-thick.

A principal endoreloid induced by a

Rel

-morphism

E

is

β

-totally bounded

iff

E

is

β

-thick.

Example

1245

.

There is a

β

-totally bounded endoreloid which is not

α

-totally

bounded.

Proof.

It follows from the example above and properties of principal en-

doreloids.

13.3. Special case of uniform spaces

Remember that

uniform space

is essentially the same as symmetric, reflexive

and transitive endoreloid.

Theorem

1246

.

Let

f

be such an endoreloid that

f

f

1

v

f

. Then

f

is

α

-totally bounded iff it is

β

-totally bounded.

Proof.

. Proved above.