proximity space

proximity space

Let

X

be a set. A

binary relation

δ

on

P

⁢

(

X

)

, the

power set

of

X

, is called a

nearness relation

on

X

if it

satisfies

the following conditions: for

A

,

B

∈

P

⁢

(

X

)

,

1.

if

A

∩

B

≠

∅

, then

A

⁢

δ

⁢

B

;

2.

if

A

⁢

δ

⁢

B

, then

A

≠

∅

and

B

≠

∅

;

3.

(

symmetry

) if

A

⁢

δ

⁢

B

, then

B

⁢

δ

⁢

A

;

4.

(

A

1

∪

A

2

)

⁢

δ

⁢

B

iff

A

1

⁢

δ

⁢

B

or

A

2

⁢

δ

⁢

B

;

5.

A

⁢

δ

′

⁢

B

implies the

existence

of

C

⊆

X

with

A

⁢

δ

′

⁢

C

and

(

X

-

C

)

⁢

δ

′

⁢

B

, where

A

⁢

δ

′

⁢

B

means

(

A

,

B

)

∉

δ

.

If

x

,

y

∈

X

and

A

⊆

X

, we write

x

⁢

δ

⁢

A

to mean

{

x

}

⁢

δ

⁢

A

, and

x

⁢

δ

⁢

y

to mean

{

x

}

⁢

δ

⁢

{

y

}

.

When

A

⁢

δ

⁢

B

, we say that

A

is

δ

-near

, or just

near

B

.

δ

is also called a

proximity relation

, or

proximity

for short. Condition 1 is

equivalent

to saying if

A

⁢

δ

′

⁢

B

, then

A

∩

B

=

∅

. Condition 4 says that if

A

is near

B

, then any

superset

of

A

is near

B

.

Conversely

, if

A

is not near

B

, then no subset of

A

is near

B

. In particular, if

x

∈

A

and

A

⁢

δ

′

⁢

B

, then

x

⁢

δ

′

⁢

B

.

Definition

. A set

X

with a proximity as defined above is called a

proximity space

.

For any subset

A

of

X

, define

A

c

=

{

x

∈

X

∣

x

⁢

δ

⁢

A

}

. Then

c

is a

closure operator

on

X

:

Proof.

Clearly

∅

c

=

∅

. Also

A

⊆

A

c

for any

A

⊆

X

. To see

A

c

⁢

c

=

A

c

, assume

x

⁢

δ

⁢

A

c

, we want to show that

x

⁢

δ

⁢

A

. If not, then there is

C

⊆

X

such that

x

⁢

δ

′

⁢

C

and

(

X

-

C

)

⁢

δ

′

⁢

A

. The second part says that if

y

∈

X

-

C

, then

y

⁢

δ

′

⁢

A

, which is equivalent to

A

c

⊆

C

. But

x

⁢

δ

′

⁢

C

, so

x

⁢

δ

′

⁢

A

c

. Finally,

x

∈

(

A

∪

B

)

c

iff

x

⁢

δ

⁢

(

A

∪

B

)

iff

x

⁢

δ

⁢

A

or

x

⁢

δ

⁢

B

iff

x

∈

A

c

or

x

∈

B

c

.∎

This turns

X

into a

topological space

. Thus any proximity space is a topological space induced by the closure operator defined above.

A proximity space is said to be

separated

if for any

x

,

y

∈

X

,

x

⁢

δ

⁢

y

implies

x

=

y

.

Examples

.

•

Let

(

X

,

d

)

be a

pseudometric space

. For any

x

∈

X

and

A

⊆

X

, define

d

⁢

(

x

,

A

)

:=

inf

y

∈

A

⁡

d

⁢

(

x

,

y

)

. Next, for

B

⊆

X

, define

d

⁢

(

A

,

B

)

:=

inf

x

∈

A

⁡

d

⁢

(

x

,

B

)

. Finally, define

A

⁢

δ

⁢

B

iff

d

⁢

(

A

,

B

)

=

0

. Then

δ

is a proximity and

(

X

,

d

)

is a proximity space as a result.

•

discrete proximity

. Let

X

be a non-empty set. For

A

,

B

⊆

X

, define

A

⁢

δ

⁢

B

iff

A

∩

B

≠

∅

. Then

δ

so defined is a proximity on

X

, and is called the

discrete proximity

on

X

.

•

indiscrete proximity

. Again,

X

is a non-empty set and

A

,

B

⊆

X

. Define

A

⁢

δ

⁢

B

iff

A

≠

∅

and

B

≠

∅

. Then

δ

is also a proximity. It is called the

indiscrete proximity

on

X

.

References

1

S. Willard,

General Topology

,
Addison-Wesley, Publishing Company, 1970.

2

S.A. Naimpally, B.D. Warrack,

Proximity Spaces

, Cambridge University Press, 1970.

Title

proximity space

Canonical name

ProximitySpace

Date of creation

2013-03-22 16:48:11

Last modified on

2013-03-22 16:48:11

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

17

Author

CWoo (3771)

Entry type

Definition

Classification

msc 54E05

Synonym

near

Synonym

proximity

Synonym

proximity relation

Defines

nearness relation

Defines

separated proximity space

Defines

discrete proximity

Defines

indiscrete proximity

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