closure space

closure space

Call a set

X

with a

closure operator

defined on it a

closure space

.

Every

topological space

is a closure space, if we define the closure operator of the space as a

function

that takes any subset to its

closure

. The

converse

is also true:

Proposition 1

.

Let

X

be a closure space with

c

the associated closure operator. Define a “

closed set

” of

X

as a subset

A

of

X

such that

A

c

=

A

, and an “

open set

” of

X

as the

complement

of some closed set of

X

. Then the

collection

T

of all open sets of

X

is a

topology

on

X

.

Proof.

Since

∅

c

=

∅

,

∅

is closed. Also,

X

⊆

X

c

and

X

c

⊆

X

imply that

X

c

=

X

, or

X

is closed. If

A

,

B

⊆

X

are closed, then

(

A

∪

B

)

c

=

A

c

∪

B

c

=

A

∪

B

is closed as well. Finally, suppose

A

i

are closed. Let

B

=

⋂

A

i

. For each

i

,

A

i

=

B

∪

A

i

, so

A

i

=

A

i

c

=

(

B

∪

A

i

)

c

=

B

c

∪

A

i

c

=

B

c

∪

A

i

. This means

B

c

⊆

A

i

, or

B

c

⊆

⋂

A

i

=

B

. But

B

⊆

B

c

by definition, so

B

=

B

c

, or that

⋂

A

i

is closed.
∎

𝒯

so defined is called the

closure topology

of

X

with respect to the closure operator

c

.

Remarks

.

1.

A closure space can be more generally defined as a set

X

together with an

operator

cl

:

P

⁢

(

X

)

→

P

⁢

(

X

)

such that

cl

satisfies

all of the Kuratowski’s

closure axioms

where the equal sign “

=

” is replaced with

set inclusion

“

⊆

”, and the preservation of

∅

is no longer assumed.

2.

Even more generally, a closure space can be defined as a set

X

and an operator

cl

on

P

⁢

(

X

)

such that

–

A

⊆

cl

⁡

(

A

)

,

–

cl

⁡

(

cl

⁡

(

A

)

)

⊆

cl

⁡

(

A

)

, and

–

cl

is

order-preserving

, i.e., if

A

⊆

B

, then

cl

⁡

(

A

)

⊆

cl

⁡

(

B

)

.

It can be easily deduced that

cl

⁡

(

A

)

∪

cl

⁡

(

B

)

⊆

cl

⁡

(

A

∪

B

)

. In general however, the

equality

fails. The three axioms above can be shown to be

equivalent

to a single axiom:

A

⊆

cl

⁡

(

B

)

iff

cl

⁡

(

A

)

⊆

cl

⁡

(

B

)

.

3.

In a closure space

X

, a subset

A

of

X

is said to be closed if

cl

⁡

(

A

)

=

A

. Let

C

⁢

(

X

)

be the set of all closed sets of

X

. It is not hard to see that if

C

⁢

(

X

)

is

closed under

∪

, then

cl

“

distributes over

”

∪

, that is, we have the equality

cl

⁡

(

A

)

∪

cl

⁡

(

B

)

=

cl

⁡

(

A

∪

B

)

.

4.

Also,

cl

⁡

(

∅

)

is the smallest closed set in

X

; it is the bottom

element

in

C

⁢

(

X

)

. This means that if there are two

disjoint

closed sets in

X

, then

cl

⁡

(

∅

)

=

∅

. This is equivalent to saying that

∅

is closed whenever there exist

A

,

B

⊆

X

such that

cl

⁡

(

A

)

∩

cl

⁡

(

B

)

=

∅

.

5.

Since the distributivity of

cl

over

∪

does not hold in general, and there is no guarantee that

cl

⁡

(

∅

)

=

∅

, a closure space under these generalized versions is a more general system than a topological space.

References

1

N. M. Martin, S. Pollard:

Closure Spaces and Logic

, Springer, (1996).

Title

closure space

Canonical name

ClosureSpace

Date of creation

2013-03-22 16:48:08

Last modified on

2013-03-22 16:48:08

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

12

Author

CWoo (3771)

Entry type

Derivation

Classification

msc 54A05

Defines

closure topology

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