proximal neighborhood

proximal neighborhood

Let

X

be a set and

P

⁢

(

X

)

its

power set

. Let

≪

be a

binary relation

on

P

⁢

(

X

)

satisfying the

following conditions, for any

A

,

B

⊆

X

:

1.

X

≪

X

,

2.

A

≪

B

implies

A

⊆

B

,

3.

A

≪

B

and

C

≪

D

imply

A

∩

C

≪

B

∩

D

,

4.

A

≪

B

implies

B

′

≪

A

′

(

′

is the

complement

operator

)

5.

A

⊆

B

≪

C

⊆

D

, then

A

≪

D

, and

6.

if

A

≪

B

, then there is

C

⊆

X

, such that

A

≪

C

≪

B

.

By 1 and 4, it is easy to see that

∅

≪

∅

. Also, 3 and 4 show that

A

∪

C

≪

B

∪

D

whenever

A

≪

B

and

C

≪

D

. So

≪

is a

topogenous order

, which means

≪

is

transitive

and anti-symmetric. Under this order relation, we say that

B

is a

proximal neighborhood

of

A

if

A

≪

B

.

The reason why we call

B

a “proximal”

neighborhood

is due to the following:

Theorem 1

.

Let

X

be a set. The following are true.

•

Let

≪

be defined as above. Define a new

relation

δ

on

P

⁢

(

X

)

:

A

⁢

δ

′

⁢

B

′

iff

A

≪

B

. Then

δ

so defined is a

proximity relation

, turning

X

into a proximity space.

•

Conversely

, let

(

X

,

δ

)

is a proximity space. Define a new relation

≪

on

P

⁢

(

X

)

:

A

≪

B

iff

A

⁢

δ

′

⁢

B

′

. Then

≪

satisfies

the six

properties

above.

Proof.

Suppose first that

X

and

≪

are defined as above. We will verify the

individual

nearness relation axioms of

δ

by proving their

contrapositives

in each case, except the last axiom:

1.

if

A

⁢

δ

′

⁢

B

, then

A

≪

B

′

, or

A

⊆

B

′

, so

A

∩

B

=

∅

;

2.

suppose either

A

=

∅

or

B

=

∅

. In either case,

A

≪

B

′

, which means

A

⁢

δ

′

⁢

B

;

3.

if

A

⁢

δ

′

⁢

B

, then

A

≪

B

′

, so

B

′′

≪

A

′

, or

B

≪

A

′

, or

B

⁢

δ

′

⁢

A

;

4.

if

A

1

⁢

δ

′

⁢

B

and

A

2

⁢

δ

′

⁢

B

, then

A

1

≪

B

and

A

2

≪

B

, so

(

A

1

∪

A

2

)

≪

B

, or

(

A

1

∪

A

2

)

⁢

δ

′

⁢

B

;

5.

if

A

⁢

δ

′

⁢

B

, then

A

≪

B

′

. So there is

D

⊆

X

with

A

≪

D

and

D

≪

B

′

. Let

C

=

D

′

. Then

A

≪

C

′

and

C

′

≪

B

′

, or

A

⁢

δ

′

⁢

C

and

C

′

⁢

δ

′

⁢

B

.

Next, suppose

(

X

,

δ

)

is a proximity space. We now verify the six properties of

≪

above.

1.

since

X

⁢

δ

′

⁢

∅

,

X

≪

∅

′

, or

X

≪

X

;

2.

suppose

A

⁢

δ

′

⁢

B

′

, then if

x

∈

A

, we have

x

⁢

δ

′

⁢

B

′

, implying

x

∩

B

′

=

∅

, or

x

∈

B

;

3.

if

A

≪

B

and

C

≪

D

, then

A

⁢

δ

′

⁢

B

′

and

C

⁢

δ

′

⁢

D

′

, which means

A

⁢

δ

′

⁢

(

B

′

∪

D

′

)

and

C

⁢

δ

′

⁢

(

B

′

∪

D

′

)

, which together imply

(

A

∩

C

)

⁢

δ

′

⁢

(

B

′

∪

D

′

)

, or

(

A

∩

C

)

⁢

δ

⁢

(

B

∩

D

)

′

, or

A

∩

C

≪

B

∩

D

;

4.

if

A

≪

B

, then

A

⁢

δ

′

⁢

B

′

, so

B

′

⁢

δ

′

⁢

A

(as

δ

is

symmetric

, so is its complement), which is the same as

B

′

⁢

δ

′

⁢

A

′′

, or

B

′

≪

A

′

;

5.

if

A

⁢

δ

⁢

D

′

, then

B

⁢

δ

⁢

C

′

(since

A

⊆

B

and

D

′

⊆

C

′

), so

B

≪

′

C

, a

contradiction

;

6.

if

A

≪

B

, then

A

⁢

δ

′

⁢

B

′

, so there is

D

⊆

X

with

A

⁢

δ

′

⁢

D

and

D

′

⁢

δ

′

⁢

B

′

. Define

C

=

D

′

, then

A

≪

C

and

C

≪

B

, as desired.

This

completes

the proof.
∎

Because of the above, we see that a proximity space can be equivalently defined using the proximal neighborhood

concept

. To emphasize its relationship with

δ

, a proximal neighborhood is also called a

δ

-neighbhorhood.

Furthermore, we have

Theorem 2

.

if

B

is a proximal neighborhood of

A

in a proximity space

(

X

,

δ

)

, then

B

is a (topological) neighborhood of

A

under the

topology

τ

⁢

(

δ

)

induced

by the proximity relation

δ

. In other words, if

A

≪

B

, then

A

⊆

B

∘

and

A

c

⊆

B

, where

∘

and

c

denote the

interior

and

closure operators

.

Proof.

Since

A

⁢

δ

′

⁢

B

′

, then

x

⁢

δ

′

⁢

B

′

whenever

x

∈

A

, which is the contrapositive of the statement:

x

∈

A

′

whenever

x

⁢

δ

⁢

B

′

, which is

equivalent

to

B

′

⁣

c

⊆

A

′

, or

A

⊆

B

∘

. Furthermore, if

x

∉

B

, then

x

∈

B

′

. But

A

⁢

δ

′

⁢

B

′

b

assumption

. This implies

x

⁢

δ

′

⁢

A

, which means

x

∉

A

c

. Therefore

A

c

⊆

B

.
∎

Remark

. However, not every

τ

⁢

(

δ

)

-neighborhood is a

δ

-neighborhood.

Title

proximal neighborhood

Canonical

name

ProximalNeighborhood

Date of creation

2013-03-22 16:58:25

Last modified on

2013-03-22 16:58:25

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

7

Author

CWoo (3771)

Entry type

Definition

Classification

msc 54E05

Synonym

proximity neighborhood

Synonym

δ

-neighborhood

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