 Proof.

Let

ϕ

is the monovalued, surjective function, which induces the reloid

f

.

We have

µ

f

1

ν

f

.

Let

F

GR

ν

. Then there exists

E

GR

µ

such that

E

ϕ

1

F

ϕ

.

Since

µ

is

β

-totally bounded, there exists a finite subset

A

of Ob

µ

such that

h

E

i

A

=

Ob

µ

.

We claim

h

F

ih

ϕ

i

A

=

Ob

ν

.

Indeed let

y

Ob

ν

be an arbitrary point. Since

ϕ

is surjective, there exists

x

Ob

µ

such that

ϕ x

=

y

. Since

h

E

i

A

=

Ob

µ

there exists

a

A

such that

a E x

and thus

a

(

ϕ

1

F

ϕ

)

x

. So

(

ϕa

;

y

) = (

ϕa

;

ϕx

)

F

. Therefore

y

∈ h

F

ih

ϕ

i

A

.

Theorem 20.

Let

µ

and

ν

are endoreloids. Let

f

is a principal C

′′

(

µ

;

ν

)

continuous, surjective

reloid. Then if

µ

is

α

-totally bounded then

ν

is also

α

-totally bounded.

Proof.

Let

ϕ

is the surjective binary relation which induces the reloid

f

.

We have

f

µ

f

1

ν

.

Let

F

GR

ν

. Then there exists

E

GR

µ

such that

ϕ

E

ϕ

1

F

.

There exists a finite cover

S

of Ob

µ

such that

[

{

A

×

A

|

A

S

} ⊆

E.

Thus

ϕ

(

S

{

A

×

A

|

A

S

}

)

ϕ

1

F

that is

S

{h

ϕ

i

A

× h

ϕ

i

A

|

A

S

} ⊆

F

.

It remains to prove that

{h

ϕ

i

A

|

A

S

}

is a cover of Ob

ν

. It is true because

ϕ

is a surjection and

S

is a cover of Ob

µ

.

A stronger statement (principality requirement removed):

Conjecture 21.

The image of a uniformly continuous entirely defined monovalued surjective reloid

from a (

α

-,

β

-)totally bounded endoreloid is also (

α

-,

β

-)totally bounded.

Can we remove the requirement to be entirely defined from the above conjecture?

Question 22.

Under which conditions it’s true that join of (

α

-,

β

-) totally bounded reloids is also

totally bounded?

We may consider also the following predicates expressing different kinds of what is intuitively is
understood as boundness. Their usefulness is unclear, but I present them for completeness.

E

GR

f

n

N

:

thick

α

(

E

n

)

E

GR

f

n

N

:

thick

β

(

E

n

)

E

GR

f

n

N

:

thick

α

(

E

0

E

n

)

E

GR

f

n

N

:

thick

β

(

E

0

E

n

)

n

N

:

totBound

α

(

f

n

)

4