background image

4.6.4. Connectors for uniform connectedness

Let’s find a connector which generates the same connectedness as the de-

scribed above uniform connectedness.

Proposition 10

x

U

: [

{

x

} ×

C

{

x

}

)

S

(

µ

)

for every generalized uniform

space

µ

= (

U

;

f

)

.

Proof

S

(

µ

) =

S

U

{

[

S

(

f

))

|

f

µ

}

. But

{

x

} ×

C

{

x

} ⊆

S

(

f

); thus [

{

x

} ×

C

{

x

}

)

[

S

(

f

)) and consequently

S

U

{

[

S

(

f

))

|

f

µ

} ⊆

[

{

x

} ×

C

{

x

}

).

Lemma 2

[

S

S

)

F

⇔ ∀

X

S

: [

X

)

F

for every collection

S

of sets and

every filter

F

.

Proof

Obvious.

Let

X

S

: [

X

)

F

that is

X

S, Y

F

:

X

Y

. Then

Y

F

:

S

S

Y

that is [

S

S

)

F

.

From the above lemma follows that

[

A

×

C

A

)

S

(

µ

U

[

A

×

C

A

))

x

A

: [

{

x

} ×

C

(

A

\ {

x

}

))

S

(

µ

U

[

A

×

C

A

))

[

{

x

} ×

C

{

x

}

)

S

(

µ

U

[

A

×

C

A

))

.

Because

x

A

: [

{

x

} ×

C

{

x

}

)

S

(

µ

U

[

A

×

C

A

)), we have

[

A

×

C

A

)

S

(

µ

U

[

A

×

C

A

))

⇔ ∀

x

A

: [

{

x

} ×

C

(

A

\ {

x

}

))

S

(

µ

U

[

A

×

A

))

Consequently

[

A

×

C

A

)

S

(

µ

U

[

A

×

C

A

))

X, Y

P

A

: (

X

Y

=

∅ ∧

X

Y

=

A

[

X

×

C

Y

)

S

(

µ

U

[

A

×

C

A

))

.

So, our sought-for connector is defined (for example) by the formula

X r Y

[

X

×

C

Y

)

S

(

µ

U

[(

X

Y

)

×

C

(

X

Y

)))

.

A

is connected regarding

µ

iff

f

µ, X, Y

P

U

: (

X

Y

=

A

X

[

f

]

Y

)

X, Y

P

U

: (

X

Y

=

A

⇒ ∀

f

µ

:

X

[

f

]

Y

). Thus

X r Y

⇔ ∀

f

µ

:

X

[

f

]

Y

⇔ ∀

f

µ

:

X

×

Y

f

6

=

(4)

is also a connector which induces uniform connectedness.

If

µ

is a uniformity,

X r Y

X δ Y

where

δ

is the proximity induced

by

µ

. Thus my definition of uniform connectedness is equivalent to traditional

definition of uniform connectedness. (See theorem 1 in [3].)

12