 Proof.

Because thick

α

(

E

)

thick

β

(

E

)

.

Proposition 13.

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

xyGR

f

is reflexive.

Thus

{

x

} × {

x

} ⊆

E

for

x

Ob

f

and thus

{{

x

} |

x

Ob

f

}

is a sought for finite cover of Ob

f

.

Obvious 14.

A principal endo-reloid induced by a

Rel

-morphism

E

is

α

-totally bounded iff

E

is

α

-thick.

A principal endo-reloid induced by a

Rel

-morphism

E

is

β

-totally bounded iff

E

is

β

-thick.

Example 15.

There is a

β

-totally bounded endoreloid which is not

α

-totally bounded.

Proof.

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

Special case of uniform spaces

Definition 16.

Uniform space

is essentially the same as symmetric, reflexive and transitive endo-

reloid.

Exercise 1.

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

or PlanetMath).

Theorem 17.

Let

f

is such a endoreloid that

f

f

1

f

. Then

f

is

α

-totally bounded iff it is

β

–totally bounded.

Proof.

.

Proved above.

.

For every

ε

GR

f

we have that

h

ε

i{

c

0

}

,

,

h

ε

i{

c

n

}

covers the space.

h

ε

i{

c

i

} × h

ε

i{

c

i

} ⊆

ε

ε

1

because for

x

∈ h

ε

i{

c

i

}

(the same as

c

i

∈ h

ε

1

i{

x

}

) we have

hh

ε

i{

c

i

} × h

ε

i{

c

i

}i{

x

}

=

h

ε

i{

c

i

} ⊆ h

ε

ih

ε

1

i{

x

}

=

h

ε

ε

1

i{

x

}

. There exists

ε

GR

f

such that

ε

ε

1

ε

because

f

f

1

f

. Thus for every

ε

we have

h

ε

i{

c

i

} × h

ε

i{

c

i

} ⊆

ε

and so

h

ε

i{

c

0

}

,

,

h

ε

i{

c

n

}

.

is a sought for finite cover.

Corollary 18.

A uniform space is

α

-totally bounded iff it is

β

-totally bounded.

Proof.

From the theorem and the definition of uniform spaces.

Relationships with other properties

Theorem 19.

Let

µ

and

ν

are endoreloids. Let

f

is a principal C

(

µ

;

ν

)

continuous, monovalued,

surjective reloid. Then if

µ

is

β

-totally bounded then

ν

is also

β

-totally bounded.

3