uniform proximity is a proximity

uniform proximity is a proximity

In this entry, we want to show that a

uniform proximity

is, as expected, a

proximity

.

First, the following

equivalent

characterizations

of a uniform proximity is useful:

Lemma 1

.

Let

X

be a

uniform space

with uniformity

U

, and

A

,

B

are subsets of

X

. Denote

U

⁢

[

A

]

the image of

A

under

U

∈

U

:

{

b

∈

X

∣

(

a

,

b

)

∈

U

⁢

for some

⁢

a

∈

A

}

.

The following are equivalent:

1.

(

A

×

B

)

∩

U

≠

∅

for all

U

∈

𝒰

2.

U

⁢

[

A

]

∩

U

⁢

[

B

]

≠

∅

for all

U

∈

𝒰

3.

U

⁢

[

A

]

∩

B

≠

∅

for all

U

∈

𝒰

If we define

A

⁢

δ

⁢

B

iff the pair

A

,

B

satisfy

any one of the above conditions for all

U

∈

𝒰

, we call

δ

the uniform proximity.

Proof.

(

1

⇒

2

) Suppose

(

a

,

b

)

∈

(

A

×

B

)

∩

U

. Then

b

∈

U

⁢

[

A

]

. Since

U

is

reflexive

,

(

b

,

b

)

∈

U

, or

b

∈

U

⁢

[

B

]

. This means

b

∈

U

⁢

[

A

]

∩

U

⁢

[

B

]

.

(

2

⇒

3

) For any

U

∈

𝒰

, we can find

V

∈

𝒰

such that

V

∘

V

⊆

U

. So

V

=

V

∘

Δ

⊆

V

∘

V

⊆

W

, where

Δ

is the

diagonal relation

(since

V

is reflexive). Set

W

=

V

∩

V

-

1

. By

assumption

, there is

c

∈

W

⁢

[

A

]

∩

W

⁢

[

B

]

(and hence

c

∈

U

⁢

[

A

]

∩

U

⁢

[

B

]

as well). This means

(

a

,

c

)

,

(

b

,

c

)

∈

W

for some

a

∈

A

and

b

∈

B

. Since

W

is

symmetric

,

(

c

,

b

)

∈

W

⊆

V

, so that

(

a

,

b

)

=

(

a

,

c

)

∘

(

c

,

b

)

∈

V

⊆

U

. This means that

b

∈

U

⁢

[

A

]

. As a result,

U

⁢

[

A

]

∩

B

≠

∅

.

(

3

⇒

1

) If

b

∈

U

⁢

[

A

]

∩

B

, then there is

a

∈

A

such that

(

a

,

b

)

∈

U

, or

(

A

×

B

)

∩

U

≠

∅

.
∎

We want to prove the following:

Proposition 1

.

The

binary relation

δ

on

P

⁢

(

X

)

defined by

A

⁢

δ

⁢

B

𝑖𝑓𝑓

(

A

×

B

)

∩

U

≠

∅

⁢

for all

⁢

U

∈

𝒰

is a proximity on

X

.

Proof.

We verify each of the axioms of a proximity relation:

1.

if

A

∩

B

≠

∅

, then

A

⁢

δ

⁢

B

:

pick

c

∈

A

∩

B

, then

(

c

,

c

)

∈

U

since the diagonal relation

Δ

⊆

U

for all

U

∈

𝒰

.

2.

if

A

⁢

δ

⁢

B

, then

A

≠

∅

and

B

≠

∅

:

If

A

⁢

δ

⁢

B

, then

(

A

×

B

)

∩

U

≠

∅

for every

U

∈

𝒰

, since no

U

is empty, there is

(

a

,

b

)

∈

U

such that

(

a

,

b

)

∈

A

×

B

, or

A

≠

∅

and

B

≠

∅

.

3.

(symmetry) if

A

⁢

δ

⁢

B

, then

B

⁢

δ

⁢

A

:

If

A

⁢

δ

⁢

B

, then there is

(

a

U

,

b

U

)

∈

(

A

×

B

)

∩

U

-

1

for every

U

∈

𝒰

, so

(

b

U

,

a

U

)

∈

U

, which implies

(

B

×

A

)

∩

U

≠

∅

, or

B

⁢

δ

⁢

A

.

4.

(

A

1

∪

A

2

)

⁢

δ

⁢

B

iff

A

1

⁢

δ

⁢

B

or

A

2

⁢

δ

⁢

B

:

Since

(

A

1

∪

A

2

)

×

B

=

(

A

1

×

B

)

∪

(

A

2

×

B

)

,

(

a

,

b

)

∈

(

(

A

1

∪

A

2

)

×

B

)

∩

U

iff

(

a

,

b

)

∈

(

(

A

1

×

B

)

∪

(

A

2

×

B

)

)

∩

U

=

(

(

A

1

×

B

)

∩

U

)

∪

(

(

A

2

×

B

)

∩

U

)

iff

(

a

,

b

)

∈

(

A

1

×

B

)

∩

U

⁢

or

⁢

(

a

,

b

)

∈

(

A

2

×

B

)

∩

U

.

5.

A

⁢

δ

′

⁢

B

implies the

existence

of

C

∈

P

⁢

(

X

)

with

A

⁢

δ

′

⁢

C

and

(

X

-

C

)

⁢

δ

′

⁢

B

, where

A

⁢

δ

′

⁢

B

means

(

A

,

B

)

∉

δ

.

First note that

δ

′

is symmetric because

δ

is. By assumption, there is

U

∈

𝒰

such that

U

⁢

[

A

]

∩

U

⁢

[

B

]

=

∅

(second equivalent characterization of uniform proximity from lemma above). Set

C

=

U

⁢

[

B

]

. Then

U

⁢

[

A

]

∩

C

=

∅

. By the third equivalent condition of uniform proximity,

A

⁢

δ

′

⁢

C

. Likewise,

U

⁢

[

B

]

∩

(

X

-

C

)

=

U

⁢

[

B

]

∩

(

X

-

U

⁢

[

B

]

)

=

∅

, so

B

⁢

δ

′

⁢

(

X

-

C

)

, or

(

X

-

C

)

⁢

δ

′

⁢

B

.

This shows that

δ

is a proximity on

X

.
∎

Title

uniform proximity is a proximity

Canonical name

UniformProximityIsAProximity

Date of creation

2013-03-22 18:07:21

Last modified on

2013-03-22 18:07:21

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

7

Author

CWoo (3771)

Entry type

Derivation

Classification

msc 54E17

Classification

msc 54E05

Classification

msc 54E15

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