background image

13.4. RELATIONSHIPS WITH OTHER PROPERTIES

234

. For every

up

f

we have that

h

GR

i

{

c

0

}

, . . . ,

h

GR

i

{

c

n

}

covers the space.

h

GR

i

{

c

i

} × h

GR

i

{

c

i

} ⊆

GR(

1

) because for

x

∈ h

GR

i

{

c

i

}

(the

same as

c

i

∈ h

GR

i

{

x

}

) we have

h

GR

i

{

c

i

} × h

GR

i

{

c

i

}

{

x

}

=

h

GR

i

{

c

i

} ⊆ h

GR

i

GR

1

{

x

}

=

GR(

1

)

{

x

}

.

For every

0

up

f

exists

up

f

such that

1

v

0

because

f

f

1

v

f

. Thus for every

0

we have

h

GR

i

{

c

i

} × h

GR

i

{

c

i

} ⊆

GR

0

and so

h

GR

i

{

c

0

}

, . . . ,

h

GR

i

{

c

n

}

is a sought for finite cover.

Corollary

1247

.

A uniform space is

α

-totally bounded iff it is

β

-totally

bounded.

Proof.

From the theorem and the definition of uniform spaces.

Thus we can say about just

totally bounded

uniform spaces (without specifying

whether it is

α

or

β

).

13.4. Relationships with other properties

Theorem

1248

.

Let

µ

and

ν

be endoreloids. Let

f

be a principal C

0

(

µ, ν

)

continuous, monovalued, surjective reloid. Then if

µ

is

β

-totally bounded then

ν

is

also

β

-totally bounded.

Proof.

Let

ϕ

be the monovalued, surjective function, which induces the

reloid

f

.

We have

µ

v

f

1

ν

f

.

Let

F

up

ν

. Then there exists

E

up

µ

such that

E

ϕ

1

F

ϕ

.

Since

µ

is

β

-totally bounded, there exists a finite typed subset

A

of Ob

µ

such

that

h

GR

E

i

A

= Ob

µ

.

We claim

h

GR

F

i

h

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

ϕx

=

y

. Since

h

GR

E

i

A

= Ob

µ

there exists

a

A

such that

a

(GR

E

)

x

and thus

a

(

ϕ

1

F

ϕ

)

x

. So (

ϕa, y

) = (

ϕa, ϕx

)

GR

F

. Therefore

y

∈ h

GR

F

i

h

ϕ

i

A

.

Theorem

1249

.

Let

µ

and

ν

be endoreloids. Let

f

be a principal C

00

(

µ, ν

)

continuous, surjective reloid. Then if

µ

is

α

-totally bounded then

ν

is also

α

-totally

bounded.

Proof.

Let

ϕ

be the surjective binary relation which induces the reloid

f

.

We have

f

µ

f

1

v

ν

.

Let

F

up

ν

. Then there exists

E

up

µ

such that

ϕ

E

ϕ

1

F

.

There exists a finite cover

S

of Ob

µ

such that

S

A

×

A

A

S

 

GR

E

.

Thus

ϕ

S

A

×

A

A

S

 

ϕ

1

GR

F

that is

S

n

h

ϕ

i

A

×h

ϕ

i

A

A

S

o

GR

F

.

It remains to prove that

n

h

ϕ

i

A

A

S

o

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

1250

.

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 conjec-

ture?