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4.6.3. Uniform connectedness

Let

µ

is a generalized uniform space.

Definition 14

I will denote

S

(

µ

) =

S

U

{

[

S

(

f

))

|

f

µ

}

.

Obvious 7

S

is a monotone function.

Definition 15

A set

A

is

(uniformly) connected

regarding

µ

iff

S

(

µ

U

[

A

×

C

A

))

[

A

×

C

A

)

.

Proposition 7

S

([

f

)) = [

S

(

f

))

for every digraph

f

.

Proof

S

([

f

)) =

S

U

{

[

S

(

g

))

|

g

[

f

)

}

=

S

U

{

[

S

(

f

))

}

= [

S

(

f

)).

Obvious 8

A set

A

is connected regarding a generalized uniform space

µ

iff

S

(

µ

U

[

A

×

C

A

)) = [

A

×

C

A

)

.

Uniform connectedness is a generalization of digraph connectedness:

Proposition 8

A set

A

is uniformly connected regarding

[

µ

)

iff it is connected

regarding

µ

(for every digraph

µ

).

Proof

S

([

µ

)

U

[

A

×

A

)) =

S

([

µ

(

A

×

C

A

))) = [

S

(

µ

(

A

×

C

A

))).

Thus

S

([

µ

)

U

[

A

×

A

)) = [

A

×

C

A

)

S

(

µ

(

A

×

C

A

)) =

A

×

C

A

.

Obvious 9

A set

A

is connected regarding a generalized uniform space

µ

iff

X

S

(

µ

U

[

A

×

A

)) :

X

A

×

C

A

.

Obvious 10

A set

A

is connected regarding a generalized uniform space

µ

iff

it is connected regarding

µ

U

[

A

×

A

)

.

Proposition 9

A set

A

is connected regarding a generalized uniform space

µ

iff

A

is connected regarding every digraph

f

µ

.

Proof

Let a set

A

is connected regarding

µ

and

f

µ

. Then [

f

)

µ

; consequently

[

f

)

U

[

A

×

C

A

)

µ

U

[

A

×

C

A

) and so

S

([

f

)

U

[

A

×

C

A

))

S

(

µ

U

[

A

×

C

A

))

[

A

×

C

A

). Thus

S

([

f

(

A

×

C

A

)))

[

A

×

C

A

); [

S

(

f

(

A

×

C

A

)))

[

A

×

C

A

);

S

(

f

(

A

×

C

A

))

A

×

C

A

that is

A

is connected regarding

f

.

S

(

µ

U

[

A

×

C

A

)) =

S

U

[

S

(

f

))

|

f

µ

U

[

A

×

C

A

)

 

=

S

U

[

S

(

g

h

))

|

g

µ, h

[

A

×

C

A

)

 

S

U

[

S

(

g

(

A

×

C

A

)))

|

g

µ

 

=

S

U

[

A

×

C

A

)

|

g

µ

 

= [

A

×

C

A

).

11